PROP. I. PROBL.

Tab. 2. Fig. 3. A E is a Body of Glass, whose Surface A B C D is the Segment of a Sphere, the Centre whereof is F. K B is a Ray of Light parallel to the Axis C F R. There being given C F the Radius of the Convexity, and B O the [Page 11] Distance of the Ray from the Axis; 'tis required to find the Point R, wherein this Ray, after one sole Refraction at the Point of Incidence B, crosses the Axis C R.

Produce K B at pleasure to H, and draw B F X at pleasure.

1. In the Right Angled Triangle B O F, we have given B O and B F, whence we may find the Angle B F O = H B F, which is the Incidence.

2. Then say, as 300 : to 193 :: (or as 14 to 9) so the Sine H I of the Incidence H B F : to the Sine Z L of the re­fracted Angle R B F; then H B F − R B F = H B R = B R F, which is the Angle of Refraction.

3. In the Triangle R B F, all the Angles and one of the Sides B F are known, whence we may find F R: Which was required.

PROP. II. PROBL.

A Plano-Convex Glass a f b Tab. 4. Fig. 1. Ta. 4. Fi. 1. being exposed to a distant Object with its Convex side towards it, and receiving the Ray cd parallel to the Axis ez, 'tis required to determin the Point of Concourse z, from these Data, f h the Glasses thick­ness, q f the Radius of the Convexity, and d g the distance of the Ray c d from the Axis e z.

The same Problem is proposed from the same Data, for the plane Side towards the Object.

But first for the Convex-Side to the Object: The Mecha­nical Construction is thus, Tab. 4. Fig. 1. Produce the Ray c d through the Glass at pleasure to m. And from d the Point of Incidence draw the Line d q to the Centre of the Con­vexity; this is the Perpendicular, and the Angle q d m the In­cidence or Inclination. On d as a [...] Centre at any distance strike the Arch m o q, in which make mn the Sine of the Angle of Incidence, to o p the Sine of the refracted Angle, as 14 to 9 (or o p 9 such parts as m n is 14, which is easily done by the Sector) the Line d k o shall be the Line in which the Ray would proceed; if the Medium of Glass were continued so far.

But because in this Line at k the Ray emerges from the plain Surface of the Glass a dense Medium into Air a rare Me­dium, from k raise k x perpendicular to the plain Surface a h b, and on k, as a Centre at any distance strike the Arch x u s; then the Angle x k u is the Incidence of the Ray passing from Glass to Air.

Making therefore t u the Sine of this Angle, to r s the Sine of the second refracted Angle from the Perpendicular as 9 to 14, the Line drawn from k through the Point s, and produced to the Axis at z, shews the Progress of the Ray in the Air af­ter its Emersion from Glass: Wherefore the Point z is deter­mined in the Axis.

[Page]

PORP. III. PROBL.

In a Double Convex-Glass ab, Tab. 7. Fig. 1. Ta. 7. Fi. 1. of equal or unequal Convexities; Let the Centres of the Spherical Sur­faces be k and q; cd a Ray falling parallel to the Axis kqz, and being Refracted at its Ingress at d, and at its Egress at i; 'Tis required to determin the Point of its Concourse with the Axis at z, from these Data, the Radii ky, qw, of the Convexities, and gd the Distance of the Point of Incidence from the Axis.

This is so easily done by Scale and Compass from what foregoes, that I shall not insist on a farther Explication there­of, but shall shew the more certain Method of tracing the Progress of this Ray by Calculation.

Produce c d directly to e, and d i to m. We have k y + q w − w y = k q.

First therefore in the Right-angled Triangle q d g, g d and q d being given, we may find the ∠ g q d = q d e and the side q g. Then k q − q g = k g.

Secondly, In the Right angled Triangle k g d, there being given k g and g d, we may have the ∠ g k d = k d c, and the side k d.

[Page 20]Thirdly, As 300 : to 193 :: so S. ∠q d e : to S. ∠ q d i.

Fourthly, q d e − q d i = e d i. And 180 − k d c − e d i = k d i.

Fifthly, In the Triangle k d i, the Angle k d i and the Sides k i, k d, being known, we have the Angle k i d = m i l. And the Angle i k d. Then g k d − i k d = q k i.

Sixthly, As 193 : to 300 :: so S. ∠ kid : to S. ∠ zil, then zil−qki=kzi.

Lastly, In the Triangle z k i, k i and all the Angles being known, we find k z, from which subtracting k y, there re­mains z y, which was required.

If by this Method we calculate the Progress of a Ray [...]through a Double Convex-Glass of equal Convexities; and the Thick­ness of the Glass be little or nothing in comparison of the Ra­dius of the Convexity; and the Distance of the Point of Inci­dence from the Axis be but small, we shall find the Point of Concourse to be distant from the Glass about the Radius of the Convexity nearly.

This our Third Proposition in most Dioptrick Writers is usually thus expressed.

Parallel Rays falling on a Double Convex-Glass of equal Convexi­ties on both sides, are united with the Axis, about the Distance of the Radius of the Convexity from the Pole of the Glass, if the Segment be but 30 Degrees.

But their Demonstrations of this Proposition are usually per­plexed enough, by reason of the smallness and multiplicity of Angles, which are expressed in their Figures, as also by the Thickness of the Glass, which they are forced to represent, and yet is to be neglected in their Demonstrations. I shall give as short a Demonstration as I can, after their Method, as follows.

Tab. 7. Fig. 2. Ta. 7. Fi. 2. Let a be the Centre of the Convexity b k l, h the Centre of the Convexity b i l, d b a Ray of Light pro­duced directly to f parallel to the Axis a h, a b e is a Right Line, and h b c is a Right Line. The first Angle of Inclina­tion [Page 21] is c b d = h b f = e b f, and the perpendicular is h b. By the first Refraction the Ray is deflected from its straight course b f and becomes b g, approaching the perpendicular b h by the Angle f b g, which is ⅓ of the Inclination e b f. Produce g b di­rectly to z. Now the Angle of Incidence on the lower Con­vexity bkl from Glass to Air is a b z = e b g, and the Perpen­dicular is a b or b e: Wherefore the Ray instead of going for­wards in b g is now, by its Emersion into Air, refracted from the Perpendicular b e, by the Angle g b h, which is therefore to be equal to ½ the second Angle of Incidence e b g. by Exper. 5. and that g b h is equal to ½ e b g, I thus demonstrate;

Which was to be Demonst.

Therefore it follows that the Ray by this second Refraction proceeds in hh, crossing the Axis in h the Centre of the Convexi­ty. Which was to be demonstrated.

See the Demonstrations of this Proposition Kepleri Dioptri. Prop. 39. Herigon. Prop. 15. Dechales Pr. 21. lib. 13. Hon. Faber Prop. 43. Sec. 8. Cherubin de Orleans la Dioptr. Oculair, Prop. 2. Zahn Telescop. Fund 2. Syntag. 1. Cap. 4. Prop. 8.

PROP. IV.

The Parallel Rays that proceed from each Collateral Point of an Object, and fall Obliquely on a Glass, are united with their Axis at the same Distance, as the Direct Parallel Rays are united with their Axis, viz. By a Plano Convex about the Distance of the Di­ameter; by a Double Equal Convex about the Distance of the Ra­dius; and by a Double Unequal Convex, as is before determin'd.

I have hitherto proposed the Properties of Glasses Proble­matically, to be Demonstrated or rather traced out by a Geo­metrical Calculation of the Rays Progress. This Method, as 'tis the most Artificial, so 'tis the most Legitimate and Exact, and wholly New and Distinct from that of other Optick Writers. I shall first observe the same Method in this Proposition, pro­posing it Problematically thus, Tab 1 [...]. Fig. 1. Tab. 10. Fig. 1. efc is a Pla­no Convex Glass, with the Convex Side towards the Object. [Page]

[Page] [Page 29] Let there be given qf the Radius of the Convexity (as in the Table after the foregoing Scholium) 20000, and fg the Glasses thickness 5858, zfqa is the Axis of the Glass it self, or of the Direct Cone of Rays, the Point of whose Concourse is in a, so that fa (as we have found in the foregoing Pro­positions) is 34112. Let rf be the Axis of an Oblique Cone of Rays, falling on the Vertex f; and making the Angle of Incidence rfz (= xdl) of any given Quantity (as in this Example suppose 14 [...]. 28'. 40''.) Let us suppose an other Ray mi Parallel to rf, and its Point of Incidence i at such a Di­stance from the Axis zfa, that being produced directly for­wards, it may pass through the Centre q of the Convexity; that is to say, let us suppose kqi to be equal to zfr. Thus I propose it for ease of Illustration, for the Calculation may as truly be performed by supposing the Ray mi at any other Distance, as I shall declare presently.

From these things given, 'tis required to find the Point l in the Axis, where the Ray mi crosses it. For the Discovery whereof we proceed thus.

Draw ik Perpendicular to zq, then in the Right-angled Triangle kqi, we have q i, and the Angle k q i to find k q and k i; then q f−f g = q g. And k q: k i :: q g:g h.

Produce r f to p, then in the Right-angled Triangle f g p, we have f g, and the Angle g f p = r f z to find the Side p g; or thus, the Triangles g f p, g q h are Similar, then g q: g h :: g f: g p. Say then, as 300: to 193 :: so S. [...] g f p: to S. [...] g f d.

In the Right-angled Triangle f g d, we have f g, and the [...] g f d, to find the Side d g, then dg + g h = d h.

Draw ho and dx Perpendiculars to ec, say then, as 193: to 300 :: so S. [...] o h q = g q h: to S. [...] o h l. But 90°− ohl= d h l. And 90° + x d l or z f r = h d l.

[Page 30]Lastly, In the Triangle hdl, we have the two Angles d h l and h d l, and the Side h d to find the Side d l; which was required.

I have said before, that the same Calculation may be per­formed for any other Ray Parallel to r f, falling at any di­stance ki from the Direct Axis z f q a; supposing with the former Data (instead of the Angle f q i) ( Tab. 10. Fig. 1.) the Distance ki (Tab. 10. Fig. 2.) be given. For then in Tab. 10. Fig. 2. T. 10. Fi. 2. draw n i b Parallel to z q, and produce q i to x, then nim = r f z. In the Right-angled Triangle q k i, having k i and q i, we get the Angle kqi = n i x, and the Side k q. And n i x−n i m = m i x = to the Angle of Incidence. And q f−q k = f k, f g−f k = k g = i b, by which in the se­veral small Triangles within the Glass, we may find the An­gle of the Rays Incidence from Glass to Air, as in the immediate foregoing Example, and so onwards in like manner.

I shall propose one Case more in this Doctrine of Oblique Rays, and that is in Tab. 11. Fig. 1. Tab. 11. Fig. 1. where the Plane Side of the Glass is turn'd towards the Object. And that we may trace a Ray in this Case by Calculation, and find out a pro­per Ray for our purpose, we are to work a little backwards, thus, q f = q h the Radius of the Convexity is given 20000, f g the thickness of the Glass is given 5858; And q f−g f= q g = 14142. Let d f be a Ray within the Glass, whose In­gress is at d, so that d g is given 1000; and its Egress is at the Glasses Vertex f: In the Right-angled Triangle d g f, we have two Sides given to find the Angle d f g. Let k d be Perpendicular to e c, and produce f d directly to x: Then say, as 193: to 300 :: so Sine of the Angle k d x = d f g: to Sine [...] k d r. And k d r−k d x = x d r. Wherefore r d is a Ray of Light falling on the Glass at d obliquely by the Angle k d r, which after its Immersion into the Glass proceeds onwards, and Emerges at the Vertex of the Glass f. This Ray at its [Page]

[Page] [Page 31] Emersion at f shall again run Parallel to r d, by what fore­goes; wherefore the Angle l f a is equal to the Angle k d r.

And thus much for the backward Operation, which we have Instituted to find a proper Ray to work with, that is, to find the Axis of an Oblique Cone, and its Angle of In­clination.

Let m i be another Ray Parallel to r d, and falling on the Glass at i, so that g i, may be given 5000. 'Tis requi­red to determine the Point l, where the Ray m i, after a Dou­ble Refraction at its Ingress at i, and at its Egress at h, meets the Axis r d f l.

Produce r d directly to p, and m i to b, and make s i Per­pendicular to ec; Seeing therefore that m i is Parallel to r d, the refracted part i h, shall be Parallel to the refracted part d f, and the Angle x d r (=p d f) shall be equal to b i h, and the Angle k d r is equal to s i m. Joyn f h.

Then in the Right-angled Triangle q g i, we have q g and g i to find the Angle g q i = q i s, and the Side q i: But q i s + s i m = m i q, and 180°− m i q = q i b. And q i b + b i h is equal to q i h.

Therefore in the Triangle q i h we have the Sides q i and q h, and the Angle q i h, to find the Angles i q h and q h i. But g q i−i q h is equal to g q h or f q h.

Then in the Isosceles Triangle f q h, we have f q and q h equal to each other, and the Angle f q h to find the Side f h, and the Angle f h q = q f h.

Produce q h directly to [...], then as 193: To 300 :: So Sine Angle q h i: To Sine of the Angle o h l. But 180°− o h l−f h q = f h l. Also 180° − q f h = h f a. And h f a + l f a = h f l.

Lastly, In the Triangle h f l, we have f h and all the Angles to find the Side f l, which was required.

PROP. V.

In Convex-Glasses, When a Nigh Object a Placed more Distant than the Focus, The Rule for Determining the Distant Base, or Respective Focus is This,

As the Difference, between the Distance of the Object, and Focus:

Is To the Focus, or Focal Length ::

So the Distance of the Object from the Glass:

To the Distance of the Respective Focus or Distinct Base from the Glass.

PROP. VI.

An Object being placed in the Focus of a Convex-Glass, the Rays from each Point thereof, after passing the Glass become Parallel;

Tab. 14. F. 4. Tab. 14. Fig. 4. ABC is an Object placed in the Focus of the Glass RS. The Rays form each Point of this Object [...]low upon the Glass, as here we have expressed those Points after passing the Glass become Parallel, those from A being a a a, from B b b b, from C c c c. For let us imagine a distant Object to send its Parallel Ray, a a a from its upper Point, b b b from its mid­dle Point, c c c from its lower Point: These (by what foregoes) shall be formed by the Glass into the distinct Base A B C. Let us now conceive this distinct Base, A B C, to be made the Object; and the Case is most plain, that its Rays must needs [...]e remitted back again in the same manner as they were re­ceived [Page] [Page]

[Page 47] from the distant Object on the side a b c, that is, Pa­rallel. For the Progress of a Ray is reciprocal. per Exp. 8. Vid. Prop. XXVII.

PROP. VII.

If an Object be placed nigher a Convex-Glass than its Focus, the Rays from each single Point thereof after they have passed the Glass, do proceed onwards diverging; But do not diverge so much as before they entred the Glass.

For if it be as much as the Glass can do, by its refractive Power to reduce to a Parallelism the diverging Rays from an Object placed in its Focus; it must needs be more than it can do, to reduce to a Parallelism the Divergency of the Rays, from an Object placed nigher to it than its Focus; The Di­vergence in this latter Posture being much greater than in the former, and consequently not so easily reduced by the refractive Power of the Glass. Tab. 15. Fig. 1. In Tab. 15. Fig. 1. cd is a Convex-Glass in whose Focus at a there is a radiating Point, from which proceed the Rays a c f, a d m, which meeting with the Glass, instead of going onwards to f and m, are re­duced thereby to c h, d k, Parallel to each other. b is another radiating Point nigher the Glass than its Focus a, we may plainly perceive the Rays b c, b d, diverge more than the Rays a c, a d. Wherefore b c, b d, instead of proceeding directly on­wards to e and n, are reduced to c g, d l; But c g and d l do yet diverge, but not so much as b c, b d.

PROP. VIII.

In Convex-Glasses when an Object is placed nigher the Glass than the Focus, the Rule for finding the Imaginary Focus is this,

As the Difference between the Distance of the Object from Glass, and the Glasses Focus:

Is to the Glasses Focus ::

So the distance of the Object from the Glass:

To the Distance of the Imaginary Focus from the Glass.

PROP. IX.

In double Convex-Glasses of equal or unequal Convexities, the Rays proceeding from the Distance of a Diameter of one Con­vexity, are united at the Distance of a Diameter of t'other Convexity.

In Tab. 15. Fig. 3. T. 15. F. 3. Let us conceive the Glass a b divided by the Pointed Right Line a b into two Plano-Convex Glasses a c b and a d b. Let the Point k be distant from the Glass a c b the Diameter of the Convexity a c b. By what foregoes, Prop. II. The Rays k a, k e, k c, k p, &c. shall be sent parallel through the Glass a c b, so that they shall become e g, c d, p q. Then falling Parallel on the Plano-Convex Glass a d b, they are unit­ed thereby in f, at the distance of its Diameter: By the fore­going Doctrine. Prop. II.

PROP. X

A Ray falling parallel to the Axis from Air, on the Concave Surface of a Glass Medium, has its Virtual Focus by its first Refraction at the Distance of a Diameter and half of the Concavity.

Tab. 15. Fig. 4. T. 15. F. 4. a b k h is a Glass Medium terminated by the Concave Surface a b, d e a Ray of Light falling parallel to the Axis g e x. I say the Ray d e, by its Refraction at the Point of Incidence e, is so refracted into e h, as if it proceeded directly from g the distance of a Diameter and half of the Con­vexity.

Let c be the Centre of the Concave-Arch a e b, draw c e di­rectly to i, produce d e directly to f, and h e directly to g. The Angle c ed = f e i is the Angle of Inclination, the Perpen­dicular is e i. By what foregoes, the Ray passing from Air a Rare Medium into Glass a Dense Medium, is refracted to­wards the Perpendicular by the Angle f e h, which must be ^1• of the Inclination f e i. Therefore f e h = d e g = e g c is half h e i = g e c. Wherefore in the Triangle c e g, the Angle c e g being double the Angle e g c, the Side c g that subtends c e g, shall be double the side c e that subtends e g c. For we suppose these Angles to be so small, that Sines and Angles, or Sides and Angles are proportional. Which was to be Demonstrated.

PROP. XI.

In Plano-Concave Glasses, when the Rays fall parallel to the Axis, the Point of Divergence or Virtual Focus is distant from the Glass the Diameter of the Concavity.

Tab. 15. Fig. 5. a b c is a Plano-Convex Glass, T. 15. F. 5. whose Ax­is e d, f g is a Ray of Light falling on this Glass parallel to its Axis. This Ray after it has passed the Glass at g, is so re­fracted into g k, that g k being produced directly backwards, it shall intersect the Axis in e, so that e b shall be the Dia­meter of the Cavity a b c.

First, Let the plain Side be turned towards the Object, as in Tab. 15. Fig. 5. Because the Ray falls perpendicular on the plain Superficies, it is not at all refracted at its Immersion into the Glass. But at g it emerges from the Concave Side of the Glass into Air; and its Inclination is g d b (d being the Centre of the Cavity a b c) and 'tis now refracted from the Perpen­dicular d g, by the Angle h g k, which is half the Inclination g d b. But h g k is equal to f g e = g e d. Wherefore in the Triangle g d e, the Angle g e d is half the Angle g d e, and therefore the Side g d is half the Side g e = e b in Glasses of large Spheres.

Secondly, Tab. 16. Fig. 1. T. 16. F. 1. Let the Concave Side e c be turned to the Ray d e. By the first Refraction the Ray suf­fers at its Immersion into Glass at e, 'tis refracted from e to f, and directed to the Point k, so that c k is thrice the Semidi­ameter n c (by the Xth hereof). Through the Point f draw i h perpendicular to the plain Side of the Glass, and con­tinue e f directly to l. The Angle of Inclination of the Ray within the Glass is h f e = i f l = d e k = e k c. Now the Ray emerging from Glass to Air, the second refracted Ray [Page] [Page]

[Page 59] ought to recede from the Perpendicular i f, so that the Angle g f i must be thrice the Angle g f l. Produce g f directly to m. Then g f i shall be equal to f m c. Wherefore in the Triangle k f m; As the Sine of the Angle k m f, or the Sine of the Angle f m c: To the Sine of the Angle f k m :: So f k: To f m, and (neglecting the Thickness of the Glass) So is k c: To m c. And Angles and Sines being proportional, k c shall be to m c as 3 to 2; but k c is Thrice the Semidia­meter, therefore m c is the Diameter. Which was to be De­monstrated.

PROP. XII.

In Double Concaves of equal Cavities, Parallel Rays have their Virtual Focus, or Point of Divergence at the Distance of the Radius of the Concavity.

Tab. 16. F. 2. ab is a Concave of equal Cavities, that is, Tab. 16. Fig. 2. m c the Radius of the Cavity e c, is equal to g n the Radius of the Cavity n i. Let the Ray d e fall thereon paral­lel to the Axis k c. This Ray, after it has suffered a [Page 60] double Refraction, shall proceed in i h, as if it came directly from m, the Centre of the Cavity.

Let c k be made three times c m, draw k e i l. By the first Refraction the Ray is propagated into e i, as if it proceeded from k (by Prop. X.). From the Centre g draw the Perpen­dicular g i f. The Angle of Inclination of the Ray within the Glass is e i f = l i g. The Ray must therefore at its Egress at i be refracted, so that the Angle of Refraction h i l = k i m must be half the Inclination l i g. Wherefore in the Triangle k i g, As the Sine of the Angle k i g, or of the Complement to 180° viz. l i g: To the Sine of the Angle at g :: So k g = 4 : To i k or n k = 3. And as the Sines, so are the Angles, seeing they are supposed very acute: Therefore, As 4 : To 3 :: So ∠lig: To the ∠ g: Add h i l = 2 to l i g = 4, we have h i g = 6. Wherefore ∠hig: Is to the Angle g :: As 6: To 3 :: or 2: To 1. But the Sine of the Angle h i g is equal to the Sine of the Angle g i m; and consequently in the Triangle g i m, As Sine of the Angle g i m: To the Sine of the Angle g :: So g m: To i m. Therefore g m: Is to i m :: As 2: To 1. So that neglecting the Thickness of the Glass, m i may be taken for the Radius of the Cavity, and g m is double thereto. Wherefore [...]tis Demonstrated, that m is the Virtual Focus. Which was to be Demonstrated.

PROP. XIII.

In double Concaves of equal or unequal Concavities, the Virtual Focus or Point of Divergence of the Parallel Rays is deter­mined by this Rule:

As the Sum of the Radii of both Concavities:

Is to the Radius of either Concavity ::

So the Double Radius of t'other Concavity:

To the Distance of the Virtual Focus.

[Page 61] Tab. 16. Fig. 3. f c is the Radius of the Concavity e c, T. 16. F. 3. make k c thrice f c, l o is the Radius of the Cavity o i; let l m be put equal to l o, and m n be put = l m, that is, o n shall then be equal to thrice o l.

I say the Ray d e, after a double Refraction, shall pro­ceed in i h, as if it came directly from g, and that it shall be, As s l the Aggregate of the Semidiameters (neglecting the Thickness of the Glass): To f c the Radius of one Conca­vity :: So o m the Diameter of t'other Concavity: To g c the Distance of the Point of Divergence.

By the first Refraction the Ray proceeds in e i, as if it came directly from k, k e i p being a Right Line ( Prop. X.).

Then the Inclination within the Glass is q i e = p i l. And the second Angle of Refraction is p i h = g i k, which is to be half the Angle of Inclination p i l.

Wherefore Sines and Angles being proportional, and the Thickness of the Glass neglected, we thus proceed in the De­monstration.

PROP. XIV.

In Concave Glasses, if the Point to which the incident Ray conver­ges, be distant from the Glass farther than the Virtual Focus of Parallel Rays, the Rule for finding the Virtual Focus of this Ray, is this:

As the difference between the distance of this Point from the Glass, and the distance of the Virtual Focus from the Glass:

Is to the distance of the Virtual Focus ::

So the distance of this Point of Convergence from the Glass:

To the distance of the Virtual Focus of this Converging Ray.

[Page 64] Tab. 17. Fig. 1. a b is a Plano Concave Glass, T. 17. F. 1. d the Centre of the Concavity, d a o a right Line, c f the length of the Vir­tual Focus, which (by what foregoes, Prop. XI.) is equal to 2 a d or 2 d c; so that f d is equal to 3 d c or 3 a d. Let n a be a Ray of Light converging directly towards the Point g. If it be made according to the Rule in the Proposition, g f : f c :: g c : ch. Then h shall be the Virtual Focus of the Ray n a, that is the Ray n a after the double Refraction it receives by the Glass shall proceed onwards in a m, as if it came directly from h, mah be­ing a Right Line.

The Proof hereof is after the same manner with the Prop. V. beforegoing; and we suppose (as in that) the Thickness of the Glass to be nothing in Comparison to the Focal Length; and likewise that the Angles of Incidence are so small, that they are as the Sines; also that the Breadth of the Glass is so little, that it makes the Angles a g c, a d c, a h c, a p c, &c. very small and Proportionable to their Sines.

PROP. XV.

In Concave Glasses a b (T. 17. F. 2.) if the Point d to which the Incident Ray c a converges, Tab. 17. Fig. 2. be nigher to the Glass than the Virtual Focus of Parallel Raysi; the Rule to find where it crosses the Axis at h, is this,

As the Excess of the Virtual Focus more than this Point of Convergency i d:

To the Virtual Focus i e ::

So the distance of this Point of Convergency from the Glass d e:

To the distance of the Point where this Ray crosses the Axis h e.

In the forementioned Figure a b is a Plano-Concave Glass, f the Center of the Cavity i e the Length of the Virtual Focus of Parallel Rays, which we know is a Diameter of the Cavi­ty, that is, = 2 f e. On this Glass there falls the Ray c a con­verging directly to the Point d. If it be made i d : i e :: d e : h e, I say h is the Point in the Axis, where the Ray c a, after its double Refraction, crosses it.

PROP. XVI. PROBL.

The Focal Lengths of two Convex-Glasses being given, the lon­ger being placed before the shorter at any given Distance less than the longer Focal Length; 'Tis required to determin the Distance of the Distinct Base from the shorter Glass.

This is solved from the VIII. Proposition hereof.

Tab. 18. Fig. 1. c x c is a Convex-Glass uniting the Paral­lel Rays, b c, b c, T. 18, F. 1 in its Focus at k; f d f is a shorter Convex uniting the Parallel Rays g f, g f, in its Focus at s. In the Problem, k x the longer Focal Length, and s d the shorter Fo­cal Length, and d x the Distance of the Glasses, are given; To find d a, the Point wherein the Rays c i, c i, are united with the Axis by the Glass f f.

By the 8th. Proposition preceding, the Focal Length d s of the Convex-Glass f f being given, and the Distance of an Ob­ject d a being given less than the Focal Length d s, to find the Distance d k of the Imaginary Focus; the Rule is, As the Dif­ference between the Distance of the Object d a, and the Focal Length d s, that is, d s − d a = a s: To the Focal Length d s :: So the Distance of the Object from the Glass a d: To the Di­stance of the Imaginary Focus from the Glass d k.

Now whereas in the 8th. Proposition d a is given, and d k sought; in this Problem d a is sought, and d k is given; for d k is equal to the Difference between the Focal Length of the longer Glass k x, and the Glass's Distance x d.

Wherefore by the foresaid 8th. Proposition, and its Corollary, it being d s − d a = a s : d s :: a d: d k. It shall be also alter­nando d s: d k :: d s −d a: a d. And Compounding d s + d k: d k :: d s: a d. But in this Problem the Three first Terms of this last Analogy are given, and the last a d required. [Page 70] Wherefore by this Analogy, the last Term a d is found. And according to the Analogy we frame this Rule for solving this Problem.

As the Focal Length of the shorter Glass + the Dif­ference between the Focal Length of the longer, and the Glasses Distance:

To the Difference between the Focal Length of the longer and the Glasses Distance ::

So the Focal Length of the shorter Glass:

To the Distance of the Distinct Base from the Glass.

PROP. XVII. PROBL.

A Convex-Glass being given, with a Concave of a larger Sphere, the Concave being placed behind the Convex, at any Distance less than the Focal Length of the Convex; 'Tis required to find the Place of the Compound Focus, or Distinct Base of these two Glasses.

This Problem is solved by the XV. Proposition hereof. Tab. 18. F. 3. T. 18. F. 3. The Convex-Glass x is placed so before the Concave-Glass e, that this Convex would unite the Parallel Rays b f, b f, in its Focus at d, and the Rays q d, q d, would cross the Axis in d, unless the Concave e were interposed. Which Con­cave is supposed to have its Virtual Focus at i more distant than d the Distance of the Focus of the Convex x.

Wherefore here is a Ray q incident on the Concave e, and converges towards a Point d, nigher the Glass e than the Vir­tual Focus of parallel Rays i; and 'tis required to determine the Point h, where it crosses the Axis. The Rule by the XV. Proposition is this,

As the Excess of the Focus of the Concave more than this Point of Convergency id:

To the Focus of the Concave ie ::

So the Distance of the Point of Convergency from the Glass de:

To the Distance of the Point where this Ray crosses the Axis he:

Now in our present Problem, the three first Terms of this Analogy are given, and the last required. For the Problem gives d x the Focal Distance of the Convex, and i e the Focus of the Concave, and e x the Distance of the Glasses. Hence we have d e (= d x − e x) the Distance from the Concave of [Page 75] the Point d, to which the Ray converges. And we have i d (= i e − d e) the Excels of the Virtual Focus of the Concave beyond the Point of Convergency d.

Wherefore by the foresaid Analogy it being, As i d: i e :: d e: he. The Rule for Solving this Problem is this,

From the Focal Length of the Convex subtract the Glasses distance. Mark the difference, then say,

As the Focal Length of the Concave—this difference:

To the Focal Length of the Concave ::

So this difference:

To the distance of the Distinct Base from the Concave.

PROP. XVIII. PROBL.

The Focal Lengths of a Convex-Glass and a Concave-Glass being given, and the Concave placed towards the Object, distant from the Convex, so that the Sum of the Concaves Focal Length and Glasses distance may be greater than the Focal Length of the Convex, and this distance of the Glasses being given also; 'Tis required to determin the distance of the Distinct Base from the Convex.

[Page 76]This is performed by the V. Proposition hereof. Tab. 18. Fig. 4. T. 18. F. 4. Let b z c be a Concave, on which there fall the pa­rallel Rays n b, m c, which ar so refracted by this Glass, that after passing it, they proceed on in b e, c f, as if they came directly from the Point a the Virtual Focus of the Concave, (by Prop. X. XI. XII. hereof) but in e and f meeting with the Convex-Glass, whose Focus is on each side it as s, s, the Rays diverge not to i and g, but are refracted, so that they cross the Axis in k.

We may therefore conceive these Rays, not as passing through the Concave b c, but as proceeding directly from the [...]igh Point a; and then to determine the Point k, where the cross the Axis, the Rule is in the Prop. V. thus,

As sa the difference between the Focus sd of the Convex-Glass ef, and the distance da of the Object from the Glass:

To the Focus sd of the Convex-Glass ::

So the distance da of the Object from the Glass:

To the distance of the respective Focus dk.

Now in this Problem, the three first Terms of this Ana­logy are given, and the fourth d k is required. For the Sum of the Concaves Focal Length and the distance of the Glasses, that is, z a + d z is equal to d a; and d a—the Convex's Focal Length that is d a − s d is equal to s a.

Wherefore we solve the Problem by this Canon.

From the Sum of the Focus of the Concave and distance of the Glasses subtract the Focus of the Convex, that is, z a + d z − s d = sa. Keep this difference, then say,

As this difference sa:

To the Focus of the Convex sd ::

So the Sum of the Concaves Focus and Glasses distance za + + dz = ad:

To the distance of the Focus from the Convex d k.

Which is required by the Problem, and is the very Ana­logy of our Prop. V. [Page]

[Page]

PROP. XIX.

In a Meniscus, if both spherical Superficies have the same Diame­ter, the Ray that falls thereon Parallel to the Axis, after its second Refraction proceeds again Parallel.

Tab. 20. F. 1. mn is a Meniscus, b c the Semidiameter of the Convexity e c, f a the Semidiameter of the Concavity i a, d e a Ray of Light Parallel to the Axis k g. I say this Ray after a double Refraction, one at e, and t'other at i, proceeds in i h Parallel to k g.

[Page 84] We here suppose the Thickness of the Glass inconsiderable in respect of the Semidiameter of the Sphere on which 'tis form'd; and therefore we wholly neglect it.

We suppose also that Sines and Angles and Sides in these small Angles are Proportional.

Let g c = g a be made Triple b c = f a, the Ray d e by its first Refraction at its Entrance into the Glass at e is refracted towards g, eig being a Right Ling; at i the Ray emerges from the Glass on the Concave Surface i a. Draw f i l. Now the Angle of Inclination within the Glass is l i e = f i g, and if the Ray be refracted into i h, the Angle of Refraction is hig = igf, which ought therefore to be ½ the Inclination f i g, and, that it is so, I thus prove.

In the Triangle g i f, As g f: To i f :: So the Angle g i f: To the Angle i g f. But g f is double i f, therefore ∠gif is double ∠igf. Which was to be Demonstrated.

PROP. XX.

In a Meniscus, if the Semidiameter of the Concavity be triple the Semidiameter of the Convexity, the Focal Length is equal to the Semidiameter of the Concavity.

This is most evident, if the Convex-side be turned to the Ray, T. 20, F. 2. as Tab. 20. Fig. 2. hi is a Ray parallel to the Axis a b, d c the Semidiameter of the Convexity f c i g, b e = 3 d c the Semidiameter of the Concavity fekg. The Ray, by the first Refraction it suffers on the Convex Surface at i, is directed to concur with the Axis at a Diameter and an half of the Convexity, that is, in b. But b is the Centre of the Concavity; wherefore the Ray i k within the Glass falls perpendicularly on the Concave Surface f k g; and therefore is not at all refra­cted by its Emersion from the Glass, but proceeds onwards directly in i k b. Which was to be Demonstrated.

[Page 85]But let the Concave-side be towards the Ray, as Tab. 20. Fig. 3. fc T. 20. F. 3. is the Semidiameter of the Conexity i c, g k = 3 f c the Semidiameter of the Convcavity e k, d e a Ray parallel to the Axis, produced directly to m. Let l c be made equal to g k, I say the Focus shall be at l, l c being equal to the Se­midiameter of the Concavity. Draw g e o. The Angle of the Rays Inclination on the Concave Surface is d e g = o e m (Ang. ad Verticem) and the Ray by the first Refraction it receives on the Concave Surface at e is so refracted as if it came from the Point h distant a Diameter and half of the Concavity then draw hein directly. Here m e n = d e h = e h g is the Angle of Re­fraction, which is therefore [...] of the Inclination d e g = e g k = oem. But we suppose, as the Angles so are the Sines, and so are the Sides. Wherefore in the Triangle e g h, s ∠ e g h = s ∠ e g k: s ∠ e h k :: e h = k h : e g, that is, ∠ e g k: ∠ e h k :: k h: e g. But the Angle e g k is equal to 3 ∠ e h k, and therefore k h is equal to 3 e g, and consequently g h is double e g or g k. From f the Centre of the Convexity draw f i p, this is perpendicular to the Convex Surface i c. Wherefore seeing g h is double g k, and g k is triple f c or f i, h f shall be octuple f i (by the Scheme). And the Second Angle of Inclination h i f on the Convex Surface shall be octuple of the Angle at h, because their Correspondent subtending Sides are as 8 to 1. But h i f is equal to p i n (Ang. ad Vert.) And because at the Rays Egress from Glass to Air, the Angle of Refraction n i l ought to be half the Inclination p i n, so let it be made; and then seeing the Angle p i n is octuple the Angle at h, its half n i l shall be quadruple the Angle at h; from the Angle n i l subtract the Angle n i m = d i h = ∠ h, there remains the Angle m i l = i l f = 3 ∠ h; wherefore the Angle at l and e g k = 3 e h k are equal. And then in the Triangle e l g (neglecting the Thickness of the Glass) e l and e g, or c l and k g the Semidiameter of the Con­cavity are equal. Which was to be Demonstrated.

PROP. XXI.

In a Meniscus, the Semidiameter of whose Convexity is triple the Semidiameter of the Concavity, The Virtual Focus is distant the Semidiameter of the Convexity.

Tab. 20. F. 4. T. 20. F. 4. a b is the Semidiameter of the Concavity b d, k c = 3 a b is the Semidiameter of the Convexity c f, r d a Ray parallel ot the Axis k c. This Ray by the first Refraction it receives on the Concave Surface at d, is refracted into d e, as if it came directly from the Point k (by Prop. X.), wherefore d e falls perpendicular on the Convex Surface (by D e f. 10.) and consequently is not refracted at its egress from the Glass (by Exp. 3.) wherefore the Virtual Focus is at k. Which was to be Demonstrated.

PROP. XXII.

In whatever Meniscus wherein the Semidiameters of the Convexity and Concavity are unequal; The General Rule that assigns the Focus either Real or Virtual is this,

As the difference of the Semidiameters:

To either of the Semidiameters, whether of the Convexity or Concavity ::

So is the Diameter, of the other Surface:

To the Focus Real or Virtual.

[Page 87] ( Case 1.) I shall demonstrate this Proposition in Three several Cases, and the First shall be, Where the Semidiameter of the Concave is greater than the Semidiameter of the Convex, but less than triple the Semidiameter of the Convex.

( Case 2.) Wherein the Semidiameter of the Concave is great­er than triple the Semidiameter of the Convex.

And in these Two Cases the Glass has a Real Focus.

( Case 3.) Wherein the Semidiameter of the Concave is less than the Semidiameter of the Convex, and then the Glass has only a Virtual Focus.

'Tis here supposed, that the Breadth of the Glass is so little, and the Angles so small, that Sines and Angles and Sides may be proportional. For an Angle, we often assume its Com­plement to 180°, as having the same Sine.

'Tis supposed likewise that the Thickness of the Glass is in­considerable, and therefore 'tis wholly neglected.

First therefore for the First Case. Tab. 20. Fig. 5. T. 20. F. 5. Let d e be a Ray parallel to the Axis c k, f c is the Semidiameter of the Convexity e c; g l is the Semidiameter of the Concavity o l. Make h c triple f c. Then by the first Refraction the Ray suf­fers at e, 'tis directed towards h (by Prop. I.). Wherefore the Angle of Inclination upon its Emersion from the Glass is g o h. Let the Angle of Refraction h o k be made half g o h. Then k is the Focus, and it being so made, I say, the difference of the Semidiameters = f g: Is to f c the one Semidiameter :: As 2 g l the other Diameter: To c k the Focal Length.

PROP. XXIII. PROBL.

The Semidiameter of the Convexity being given, 'tis required to find the Semidiameter of the Concavity, that a Meniscus formed with this Convexity and Concavity may unite the Parallel Rays at a given distance. Vid. Barrow Lect. Opt. 14. pag. 102. &c. Edit. Lond. 1672. 4 to.

Tab. 21. F. 2. T. 21. F. 2. a e = a d is the given Semidiameter of the Con­vexity, d f is the given Distance, at which the Parallel Rays are to be united, z e produced to k is a Ray Parallel to the Axis d f. Draw a e h on the Point of Incidence e as a Centre at any Interval, strike the Arch k o a, and make the sine of the Angle of Incidence k m, to the Sine of the first refracted Angle o n, as 300 to 193, or as 14 to 9. Draw e o l produced the other way to c. The Ray by the first Refraction is directed to l; in the Line l e take any Point at Pleasure i, so that e i may be the Thickness of the Glass which is to be for­med; which Thickness is here represented considerable, for Illustration sake, but it being really of no Moment in Glasses of small Segments of large Spheres, 'tis neglected altogether in Demonstration. Draw f i which we suppose equal to f d. And make the Angle l i q: To the Angle l i f :: As 193: To 107. Draw q i which cuts the Azis in b. On b as a Centre strike the Arch of the Concavity ix. I say bi = bx is the Semidiameter of the Concavity, which added on t'other side the Glass to the former given Convexity, shall unite the Parallel Rays at the given distance d f.

PROP. XXIV.

An intire Glass-Sphere Unites the Parallel Rays at the Distance almost of half its Semidiameter behind it.

'Tis here supposed that the Incident Rays do not fall di­stant from the Axis more than 20 or 30 Degrees.

[Page 94] Tab. 21. f. 3. T. 21. F. 3. a k is a Glass-Sphere. On the Point d there falls the Ray b d Parallel to the Axis a k. I say this Ray after a Double Refraction concurs with the Axis in the Point f, so that f k is almost half the Semidiameter of the Sphere.

c is the Centre of the Sphere, make k h equal to the Semidiameter c k, then a h is a Diameter and Half. And the Ray b d by the First Refraction proceeds in d e direct­ly towards the Point h. But at e it emerges from Glass to Air. Draw the Perpendicular c e g, the Angle of this second Inclination is c e d = g e h. Now the Ray instead of proceeding directly to h is refracted from the Perpendicu­lar e g by the Angle f e h, which is therefore to be half c e d. And that it is so, I thus prove. c e d is equal to e c h + e h c (32. 1. Eucl.) but e c h and e h c are to sense equal, their Subtenses e h and e c being very near equal : wherefore e h c is half c e d. But e h c is equal to f e h (their Subtenses f e and f h being as to sense equal; for f k by supposition is equal to f h, and f e is very near equal to f k, therefore f e is almost equal to f h) and consequently the Angle f e h is half c e d or g e h. Which was to be De­monstrated.

PROP. XXV.

A Glass Hemisphere Unites the Parallel Rays at the Distance of a Diameter and one third of a Semidiameter from the Pole of the Glass.

Tab. 21. f. 4. T. 21. F. 4. a d c is a Glass Hemisphere, b d a Ray of Light Parallel to the Axis a h. This Ray after a Double Refraction Crosses the Axis in the Point f, so that f c is a Semidiameter and one third of a Semidiameter. Make c k [Page] [Page]

[Page 95] and k h each equal to c a. By the first Refraction at d, the Ray is directed towards h in d e h. At e the Point where it emerges from the Glass, draw p e r Perpendicular to the Surface c e. Now the second Angle of Inclination is d e p = h e r (15. 1.) = k h e (29. 1.) = k e h. (For k e being nighly equal to k c, 'tis also nighly equal to k h; Then in the Equilateral Triangle k e h, the Angle k e h shall be nigh­ly equal to the Angle k h e) And the Angle of Refraction is f e h. And this we are to prove equal to [...] the Inclination h e r or e h f. Seeing k h is equal to c k, and k f is ⅓ of k h by Construction, it follows that c f is double f h (for c k + k f = c f = ³³ + ⅓ = ⁴³, and f h = ⅔, but ⁴³ is = to dou­ble ⅔) But e f is nighly equal to c f; therefore e f is nigh­ly double f h. Then in the Triangle f e h, as the Sides, so the Angles (by supposition) therefore the Angle e h f is nighly double the Angle f e h. Which was to be Demon­strated.

PROP. XXVI.

As the Distance of the Object from the Glass:

To the Distance of the Image from the Glass ::

So the Diameter of the Objects Magnitude:

To the Diameter of the Image.

Tab. 22. f. 1. a b c T. 22. F. 1. is an Object, k d m a Convex Glass in the Hole of a Dark Chamber, e f g the Image formed by this Glass on a White Paper in the Distinct Base. I say, [Page 96] As c d the Distance of the Object from the Glass: To e d the Distance of the Image from the Glass :: So a c the Objects Magnitude: To g e the Magnitude of the Image.

To shew this, we are to remember the Premises to the IV Proposition. For by them we shall find, that the Axes a d, and c d of the Luminous Cones k a m, k c m, may be consider'd as passing the Glass unrefracted. Because that after they have passed the Glass and become d g, d e, they proceed Parallel to their Course before they entred the Glass. So that in Glasses of large Spheres, and small Segments, the thickness of the Glass being inconsiderable in respect of the Focal Distance, we may neglect it. And then we have here two Similar Triangles a d c, g d e; For [...] a d c is equal to [...] e d g (15. 1. Eucl.) and the Side e g is Parallel to the Side a c (by supposition) and consequently the Angle c a d is equal to the Angle e g d. (29. 1.) and a c d is equal to the Angle g e d. Wherefore it shall be c d: e d :: a c: g e. Which was to be Demonstrated.

Note. From hence appears the Mistake committed by Monsieur Comiers in De Blegny's Zodiacus Medico-Gallicus. An. 3tio. pag 117. Who says, Diameter Imaginis Solis in Distin­ctâ Basi obtinebit ad minimum in sua Diametro Funem seu Chor­dam Dimidii Gradus magni Circuli Sphaerae, cujus Vitrum est seg­mentum. Whereas it should be, Magni Circuli Sphaerae, cujus Radius est Distantia Focalis Vitri.

What is here Demonstrated concerning the Real Image of a Convex Glass may be accommodated to the Virtual Image of a Concave.

PROP. XXVII.

The Object and its Image in the Distinct Base are Reciprocal. vid. Prop. VI.

Tab. 22. Fig. 1. T. 22. F. 1. a b c being an Object, and the Convex Glass k m Representing the Image thereof in the Distinct Base e f g. Let us now conceive e f g an Object, I say the same Glass k m the same way posited shall project the Image there­of in the Distinct Base at a b c.

This does necessarily follow from the precedent Proposi­tion, and from Exp. 8. And therefore needs no farther De­monstration.

PROP. XXVIII.

The manner of Plain Vision with the naked Eye is expounded.

Tab. 25. F. 1. T. 25. F. 1▪ a b c is an Object, i k l e m is the Globe of the Eye, furnish'd with all its Coats and Humors; But in this Figure we have only expressed the Crystalline Humour g o h, as being Principally concern'd in Forming the Image on the Fund of the Eye.

(1) From each Point in the Object we may Conceive Rays flowing on the Pupil of the Eye i k; as here from the middle Point b, there proceed the Rays b g, b o, b h; These by means of the Coats and Humours of the Eye, and especially by the Chrystalline Humour g h, are refracted and brought to­gether on the Ratina or Fund of the Eye in the Point e, and there the Point b is represented. For we may conceive the Cry­stalline Humour g h as it were a Convex-glass, in the Hole a Dark Chamber i l m k, and that d e f is the Distinct Base of this Glass. What is here said of the Point b, and its Re­presentation at e, may be understood of all the other Points in the Object, as of a and c and their Representations at f and d.

(2) And as in a dark Chamber, that has a Hole furnish'd with a Convex-glass, if the Paper, that is to receive the Image in the Distinct Base, be either nigher to, or farther from the Glass, than its due distance, the Representation thereon is con­fused; For then the Radious Pencils do not exactly determine with their Apices on the Paper; But those from one Point are mixt and confused with those from the Adjacent Points: so in [Page 104] the Case of Plain Vision, 'tis requisite that the Pencils should exactly determine their Apices at d, e, f, on the Retina, or else Vision is not Distinct.

'Tis therefore contrived by the Most Wise and Omnipotent Fra­mer of the Eye, That it should have a Power of adapting it self in some Measure to Nigh and Distant Objects. For they require different Conformations of the Eye; Because the Rays proceeding from the Luminous Points of Nigh Objects do more Diverge, than those from more Remote Objects.

But whether this variety of Conformation consist in the Cry­stallines approaching nigher to, or removing farther from the Retina; Or in the Crystallines assuming a different Convexity, sometimes greater, sometimes less, according as is requi [...]ite, I leave to the scrutiny of others, and particularly of the curious Anatomist. This only I can say, that either of these Methods will serve to explain the various Phaenomena of the Eye; And I am apt to believe, that both these may attend each other, viz. a Less Convex Crystalline requires an Elongation of the Eye, and a more Convex Crystalline requires a shortning thereof; As a more [...]Flat [...]Convex-Object-glass or of a Larger Sphere re­quires a Longer Tube, and one more Protuberant, bulging or of a smaller Sphere requires a shorter Tube.

(3) By the foremention'd Scheme we perceive, the Rays from each Point of the Object are all confused together on the Pupil in g h, so that the Eye is placed in the Point of the Greatest Confusion: But by means of the Humors and Coats thereof each Cone of Rays is separated, and brought by it self to determine in its proper Point on the Retina, there Painting distinctly the Vivid Representation of the Object; Which Representation is there perceived by the sensitive Soul (whatever it be) the manner of whose Actions and Passions, He only knows who Created and Preserves it, Whose Ways are [Page 105] Past finding out, and by us unsearchable. But of this Moral truth we may be assured, That He that made the Eye shall see.

(4) We are likewise to observe, that the Representation of the Object a b c on the Fund of the Eye f e d is Inverted. For so likewise it is on the Paper in a dark Room; there being no other way for the Radious Cones to enter the Eye or the dark Chamber, but by their Axes a o, b o, c o, crossing in the Pole o of the Crystalline or Glass. And here it may be enquired; How then comes it to pass that the Eye sees the Ob­ject Erect? But this Quaery seems to encroach too nigh the en­quiry into the manner of the Visive Faculties Perception; For 'tis not properly the Eye that sees, it is only the Organ or In­strument, 'tis the Soul that sees by means of the Eye. To en­quire then, how it comes to pass, that the Soul perceives the Object Erect by means of an Inverted Image, is to enquire in­to the Souls Faculties; which is not the proper subject of this Discourse. But yet that in this Matter we may offer at som­thing, I say, Erect and Inverted are only Terms of Relation to Up and Down, or Farther from and Nigher to the Centre of the Earth, in parts of the same thing: And that that is an Erect Object, makes an Inverted Image in the Eye, and an In­verted Object makes an Erect Image; That is, that part of the Object which is farthest from the Centre of the Earth is Painted on a Part of the Eye Nigher the Centre of the Earth, than the other parts of the Image. But the Eye or Visive Fa­culty takes no Notice of the Internal Posture of its own Parts, but uses them as an Instrument only, contrived by Nature for the Exercise of such a Faculty.

But to come yet a little nigher this difficulty; This enquiry results briefly to no more than this, How comes it to pass, that the Eye receiving the Representation of a Part of an Object on that part of its Fund which is Lowermost or nighest the Centre of the Earth, perceives that part of the Object as Upper­most [Page 106] or farthest from the Centre of the Earth? And in answer to this, let us imagine, that the Eye in the Point f receives an Impulse or Stroke by the Protrusion forwards of the Luminous Axis a o f, from the Point of the Ob­ject a; Must not the Visive Faculty be necessarily directed hereby to consider this stroke, as coming from the Top a, ra­ther than from the bottom c, and consequently should be dire­cted to conclude f the Representation of the Top?

Hereof we may be satisfy'd by supposing a Man standing on his Head: For here, tho the Upper Parts of Objects are painted on the Upper Parts of the Eye, yet the Objects are judged to be Erect. And from this Posture of a Man, the Reason ap­pears, why we have used the Words Farthest from, and Nighest to the Centre of the Earth, rather than Upper and Lower. For in this Posture, because the Upper Parts of the Object are painted on that part of the Eye nighest the Earth, (though really the upper Part of the Eye) they are judged to be farthest removed from the Earth.

What is said of Erect and Reverse may be understood of Si­nister and Dexter: But of these Physical Conjectures enough.

(5) The Image of an Erect Object being Represented on the Fund of the Eye Inverted, and yet the sensitive Faculty judging the Object Erect; it follows that when the Image of an Erect Object is Painted on the Fund of the Eye Erect, the sense Judges that Object to be Inverted.

This is a necessary Conclusion, and is of consequence for explaining some Particulars that follow.

(6) The Magnitude of an Object is Estimated by the Angle the Object subtends before the Eye: Thus in the same fore­mention [...]d Tab. 25. Fig. 1. T. 25. F. 1 the Length of the Object a c is estimated by the Angle a o c = f o d, this we call T [...]e Optic Angle.

[Page 107] From hence, & from Prop. XXVI, XXVII. it follows, that if the Eye were placed instead of the Glass at d (T. 22. F. 1.) T. 22. F. 1 and a b c or e f g were Objects, the Eye would perceive them of Equal Bigness.

I know very well the Point o, which is the Vertex of the Op­tick Angle, is variously assigned by various Authors; some pla­cing it in the Centre of the Eye; Others in the Vertex of the Crystalline; Others in the Vertex of the outward Coat or Cor­nea of the Eye: But 'tis a Matter of no great consequence, where­ever we place it; for according to the Bigness of this Angle a o c, the Image on the Fund of the Eye is Bigger or Less. Vid. Tacquet Optica. Lib. 1. Def. 4. and Prop. 3.

I know likewise, that by a curious Experiment in Opticks discover'd by an Ingenious French-Man Monsieur Mariotte, 'tis controverted, whether the Retina or Choroide be the Seat of Vi­sion, or the Place on which the Pictures of outward Objects are expressed ( vid. Philosoph. Transact. Num. 35. and 59.) But to our business it matters not which of them we pitch on; and therefore I chuse to speak as commonly 'tis presumed; and mention the Retina, or rather the Fund of the Eye, as the Place that receives this Picture.

(7) In the same ( Tab. 25. Fig. 1.) T. 25. F. 1. We perceive the Rays that flow from the Point b do proceed to the Eye Diverging, as b g, b o, b h; And if the Object a c were infinitely diitant from the Eye, or so distant from the Eye, that the Breadth of the Pupil i k were insensible in Comparison to this Distance, then the Rays b g, b o, b h, would proceed as it were Parallel, and so fall on the Eye: In both which Cases, by means of the Refractions in the Eye, they are brought together, and paint the Image of the Point b on the Fund of the Eye at e. Vid. Gregorii Opt. Prom. Pr. 30.

But in Tab. 25. Fig. 2. T. 25. F. [...]. If the Diverging Rays b v, b x, that flow from the Point b, meet the Convex Glass v x, and are thereby made to converge as v i, x k, and so fall on the Eye, [Page 108] and there passing through the Crystalline g h, are made to Converge yet more as i e, k e; Here they cross in the Point e, before they reach the Retina r t, and consequently do paint thereon the Image of the Point b confusedly, for 'tis Painted on the space r t; whereas to cause distinct Vision, it should only be painted on a correspondent Point on the Retina.

And this is the Fault of their Eyes, who are called Myopes, Purblind, or Short-sighted. For in them the Crystalline is too Convex (as in this Tab. 25. Fig. 2. both the Convex Glass and Crystalline joyn'd together make too great a Convexity) uniting the Rays before they arrive at the Retina. And there­fore they are helped by Concave-glasses, which take off from the too great Convexity of their Crystalline some part of its Refractive Power: Or rather these Concaves make the Rays Diverge so, that their Crystalline shall be sufficient only to bring them again together, so that they be not united, till they arrive at the Fund of the Eye.

Myopes are also helped by holding the Object very near; for then the Rays that fall on their Eye from any single Point do more Diverge, than when the Eye is farther from the Point, and consequently their too Convex Crystalline does but suffice to bring them together on the Retina.

(8) On the contrary, the Eyes of Old Men have their Cry­stalline too Flat (as Tab. 25. Fig. 3.) T. 25. F. 3. and cannot correct the Di­vergence of the Rays b i, b k, to make them meet on the Reti­na rt, but beyond the Eye at e. Wherefore for their Help 'tis requisite they add the Adventitious Convexity of a Glass; that both it and the Crystalline together, may be sufficient to unite the Rays just at the Retina: And from hence it appears, that Spectacles help Old Men, not by magnifying an Object, but by making its Appearance Distinct; for Old Men cannot read the largest Print without Spectacles, and yet with Spe­ctacles, they read the smallest, though these with Specta­cles [Page 109] do not appear so large, as those without Spectacles.

(9) What is said of the Confused or Distinct Representation of a Point in the Object, may be understood of the Confused or Distinct Representation of the w [...]ole Object; at least for those Parts that lye pretty nigh adjacent to that Point that is looked at. For here we do not take a Point in the strict sense of the Mathematicians, but in a Physical Sense, for the smallest Part imaginable; or as we have assumed it in the first supposition. And the whole Object consisting of such Points, what is shewn of one Point may be understood of every Point in the Object, that is, of the whole Object.

(10) 'Tis Requisite also (before I proceed farther) to ex­plain, what I mean by Clear Vision, and Faint Vision; Distinct Vision, and Confused Vision.

By what foregoes, I suppose these two latter Terms are pret­ty well understood, viz. Distinct Vision is then caused, when the Pencils of Rays from each Point of an Object do accurate­ly determine in Correspondent Points of the Image on the Re­tina: Confused Vision on the contrary, when these Pencils do intermix one with another.

But Clear Vision is only caused by a Great Quantity of Rays in the same Pencil, illuminating the Correspondent Points of the Image strongly and vigorously.

Faint Vision is then when a Few Rays make up one Pencil; And tho this may be Distinct, yet 'tis Dark and Obscure, at least not so Bright and Strong, as if more Rays concurr'd.

PROP. XXIX.

An Object seen through a Plain Glass whose Surfaces are Pa­rallel is Magnify'd thereby.

This Proposition being directly contradictory to what HONORATUS FABER asserts in the XLIII. Prop. Sec. 2. of his Synopsis Optica, I shall mark my Tab. 26. Fig. 1. T. 26. F. 1. with the same Letters wherewith he marks his 89th Fig. He ac­knowledges that the Apparent Place of an Object is changed by a Plain Glass; but not that the Visual Angle is alter'd there­by. But I shall shew that the Optic [...] Angle is Magnify'd thus; Let m l be a Plain Glass, a b an Object, a f a Ray pro­duced [Page] [Page]

[Page 111] directly to g, but Refracted into f i; And at its Emer­sion from the Glass at i, Refracted again into i h, parallel to a f g. (by Experim. 1, 2.) Produce i h directly towards k; let the Eye be at h, draw h a. The Angle, under which the Ob­ject a b appears through the Glass m l to the Eye at h, is i h b, or k h b, which is certainly greater than a h b, which is the na­tural Optick Angle. Wherefore the Eye at h through the Glass sees the Object a b, under a greater Angle than it would do without the Glass, and consequently the Object is magnify [...]d by the Glass. Which was to be Demonstrated.

But that which deceived FABER is, that because the An­gle a g b is equal to the Angle k h b, therefore (says he) the Optick Angle with and without the Glass are the same; or (as he has it) the Eye at g without the Glass would see the Object a b under the Angle a g b, and through the Glass the Object is seen under the Angle k h b equal to a g b: Which is grant­ed him. But this does only prove, that the naked Eye at g, and the Eye through the Glass at h, sees the Object under the same Angle. Whereas he should have proved (if it were possible) that the naked Eye at h, and the Eye through the Glass at h, sees the Objects under the same Angle; And then indeed he had rightly proved what he proposed, viz. that the Optick Angle is not magnify'd by the Interposition of the Glass: For 'tis not Mathematical, first to place the Eye at g without the Glass, and then at h with the Glass, and proving the Optick Angles the same, to conclude that therefore the Plain Glass does not magnifie. For at that rate one may shew that even a Convex Glass does not magnifie; For at one Station an Object shall appear under as great an Angle without it, as at another Sta­tion through the Glass.

The Reason, why the Common Window Glass, &c. through which we look, causes no sensible Alteration of O [...]jects, is be­cause 'tis too Thin. Vid. Barrow Lect. Opt. 15.

[Page 112] Concerning the Apparent Place of an Object through a Plain Glass, more hereafter.

PROP. XXX.

All Objects seen Erect through Convex Glasses are Magni­fy'd thereby.

Tab. 26. Fig. 2. T. 26. F. 2. Let a x b be an Object, o the Eye, draw a e o, b d o, directly strait from the extremities of the Object to the Eye. The Angle comprised by the Direct Rays aeo, bdo, that is the Angle a o b, is the Natural Optick Angle. Let now the Glass c f be interposed; here because the Rays a e, b d, would naturally and by their Direct course concur at o, now the Glass is interposed, their Concourse shall be acce­lerated before they arrive at o. Wherefore the Eye at o shall not perceive the extremities of the Object through the Glass by the Rays a e, b d, but by some other Rays. And these other Rays must fall either without (that is, farther from the Axis o x, than) a e, b d, or they must fall within a e, b d: But they cannot fall within a e, b d; for if a e, b d themselves be made by the Glass to concur before they arrive at o, much more shall any other Rays that fall within a e, b d, be made to concur before they arrive there: And consequently they cannot convey the Appearance of the Extreme Points a, b, to the Eye at o. Wherefore it remains, that the Rays that do this must fall with­out a e, b d: Let these Rays be a c, b f, which by the Refra­ctive Power of the Glass are bent from their Direct Course and are made to proceed in c o, f o, crossing at the Eye in o; Here the Optick Angle through the Glass is c o f, which is greater than the Natural Optick Angle a o b, and consequently the Ob­ject is Magnifyd. Which was to be Demonstrated.

PROP. XXXI.

Concerning the Apparent Place of Objects seen through Convex-glasses.

(1) In Plain Vision the Estimate we make of the Distance. of Objects (especially when so far removed, that the Inter­val between our two Eyes, bears no sensible Proportion there­to; or when look'd upon with one Eye only) is rather the Act of our Iudgment, than of Sense; and acquired by Exercise and a Faculty of comparing, rather than Natural. For Distance of it self, is not to be perceived; for 'tis a Line (or a Length) presented to our Eye with its End towards us, which must therefore be only a Point, and that is Invisible. Wherefore Distance is chiefly perceived by means of Interjacent Bodies, as by the Earth, Mountains, Hills, Fields, Trees, Houses, &c. Or by the Estimate we make of the Comparative Magnitude of Bodies, or of their Faint Colours, &c. These I say are the Chief Means of apprehending the Distance of Objects, that are con­siderably Remote. But as to nigh Objects, to whose Distance the Interval of the Eyes bears a sensible Proportion, their Di­stance is perceived by the turn of the Eyes, or by the Angle of the Optick Axes. ( Gregorii Opt. Promot. Prop. XXVIII.) This was the Opinion of the Antients, Alhazen, Vitellio, &c. And tho the Ingenious Jesuit Tacquet (Opt. Lib. I. Prop. II.) disapprove thereof, and Objects against it a New Notion of Gassendus (of a Man's seeing only with one Eye at a Time one and the same Object) yet this Notion of Gassendus being absolutely False (as I could Demonstrate, were it not beside my Present Purpose, but I refer to the 7th Chap. of the 2d Part.) it makes nothing against this Opinion.

(2) Wherefore Distance being only a Line, and not of it self perceivable; if an Object were convey'd to the Eye by [Page 114] one single Ray only, there were no other means of judging of its Distance, but by some of those hinted before. Therefore when we estimate the Distance of nigh Objects, either we take the help of both Eyes, or else we consider the Pupil of one Eye as having Breadth, and receiving a Parcel of Rays from each Radiating Point. And according to the various Inclina­tion of the Rays from one Point, on the various Parts of the Pupil, we make our Estimate of the Distance of the Object: And therefore (as is said before) by one single Eye we can only Judge of the Distance of such Objects, to whose Di­stance the Breadth of the Pupil has a sensible Proportion. To illustrate all this by Tab. 26. Fig. 3. T. 26. F. 3. Let a be a Radiating Point, sending forth the Rays a d, a e, a f, a g, a h, with all the intermediate Rays. Let p u be the Breadth of the Pupil, and first placed at c, there receiving only a e, a f, a g: Let p u be translated to b, where it receives all the Rays; and 'tis manifest that the Rays from a are differently inclined on the Pupil in one and t'other Posture, for the Rays, which fall on the Pu­pil when placed at b, diverge very much more than those that fall upon it, when placed at c: And therefore the Eye, or Vi­sual Faculty will apprehend the Distance of a from b and c to be Different. For it is observed before ( Prop. 29. Sect. 2. see also, Gregorii Opt. Promot. Prop. XXIX.) that for viewing Ob­jects Remote and Nigh, there are Requisite Various Conforma­tions of the Eye: The Rays from Nigh Objects, that fall on the Eye, Diverging more than those from more Remote Ob­jects.

(4) If therefore by Refraction through Glasses, that parcel of Rays which falls on the Pupil from each Point in Nigh Ob­jects be made to flow as close together as those from Distant Objects; or the Rays from Distant Objects be made to Di­verge, as much as if they flow'd from Nigh Objects, the Eye through such Glasses shall perceive the Place of the Object chan­ged.

[Page 115] (5) But first for a sensible and common Experiment, to shew the Change of an Objects Place by Refraction. Tab. 26. F. 4. T. 26. F. 4. represents a Vessel, on whose Bottom at a there is laid a piece of Mony, or any other remarkable Object, so that the Eye at o may just perceive the Mony over the Edge c of the Vessel. The Mony now appears to the Eye o by the direct Line a c o. Let now the Vessel be fill [...]d with Water up to g f h; let the Ray a f proceed from the Mony, and draw p f perpendi­cular to g h; the Ray a f, instead of going onwards directly to d, emerging from Water, a dense Medium to Air, deflects from the perpendicular p f; and becomes (suppose) f o. Wherefore the Eye o, now the Vessel has Water in it, sees the Silver, not by the direct Ray a c o (for that is bent from it, and escapes it) but by the refracted Ray a f o. Produce o f directly to b, the Mony shall appear as if it were at b. For the Eye is not sensible of the bending of the Ray, but is affected by it, as if it were directly strait.

(6) In like manner Tab. 26. F. 5. T. 26. F. 5. if the Point c send its Ray c e obliquely on the plain Glass a b; and after a double Re­fraction it arrive at the Eye o; This Point c is not now seen in its own proper Place, but somewhere in the Ray o g produ­ced, as at f. For the Eye is not sensible of the outward acci­dental Refraction, that attends the Ray at its passage through the Glass; but is directed by the next immediate Ray o g that falls upon it, and considers it as strait, and coming directly from the Point f.

I say moreover, that by a plain Glass, the place of an Ob­ject is changed, and brought nigher the Eye. Tab. 26. F. 6. T. 26. F. 6. c is an Object sending its Rays c d, c e, upon the plain Glass a b; these after a double Refraction proceed (by supposition) in g i, h k. Produce these directly towards f; and suppose the Pupil of the Eye large enough to receive the Rays g i, h k; the Point c shall appear to the Eye as at f.

[Page 116] (7) To determine the Locus Apparens of an Object placed nigher a Convex-glass than its Focus.

To perform this, we are to have, the Power of the Glass, the Distance of the Object from the Glass, and the Length of the Object given. Tab. 27. Fig. 1. T. 27. F. 1. a b c is an Object, whose Distance from the Glass b z is given. Let us suppose the Glass of the least thickness imaginable, that we may not be at the trouble of considering the Optick Angle, both at the Immersion and Emersion. Let the middle Point b Radiate upon the Glass, and after the Rays have passed the Glass, let them be so Refracted, as if they came directly from the Point e. From the foremention'd Data, this Point e, being the Imaginary Focus, is easily determin'd by Prop. VIII. hereof: Through e draw the Infinite Right Line d e f Parallel to a b c. Wherefore as the Locus Apparens of the Point b is at e, so the Loci of the Collateral Points a, c, shall be somewhere in the Line d e f (unless perhaps Convexes on very small Spheres, will Represent the Object Crooked or Bowed, but of this we shall take no notice) to determine which, from the Vertex of the Glass z draw the Lines z a d, z c f (or if the Glass have any Thickness, as in Tab. 27. Fig. 2. T. 27. F. 2. from the Vertex of Immersion, or outward Vertex, draw z a, z c; And from the Vertex of Emersion or Inward Vertex, draw z d, z f, Parallel to z a, z c, and then the Thickness of the Glass must be one of the Data. Vid. Prop. 47. Gregorii Opt. Promot.) the Points d and f are the Loci Apparentes of the Points a and c. For certainly were the Eye behind the Glass just at z, and Object a b c would appear under the Angle a z c, and the Point b would appear as at e, and therefore the Points a and c would appear somewhere, so as to make the Object keep the same Angle; But that can only be by the Points a and c appearing at d and f (supposing the Object to appear in its own Natural strait shape) Wherefore d and f [Page]

[Page] [...] [Page 117] are the Loci of the Points a and c, and d e f is the Locus of the Object a b c.

Hence it appears, that a Convex-Glass Represents Objects as farther off than really they are: And this is the Reason, why Pieces of Perspective (as of Churches and Long Porti­coes) appear very Natural and strong through Convex-Glas­ses duly apply'd. For these Glasses making Objects appear further off than really they are, must consequently make the Parts of the Perspective seem really Hollow'd or sunk in, the French term it Renfonce.

Hence also 'tis Manifest, why Convex-Glasses help the Eyes of those, that see only Distant Objects, as Old Men, for these make Nigh Objects appear as Distant. Vid. Prop. XXVIII. Sec. 8.

We may perceive also that the Distance between the Ob­ject and Glass continuing the same, the Locus Apparens is ne­ver alter'd, though the Eye be removed to and from the Glass. Gregorii Opt. Promot. Corol. Prop. XLVII. For the Di­stance between the Object and Glass continuing the same, the Imaginary Focus e shall always be the same.

(8) The Locus of an Object expos'd to a Convex-Glass in its Fo­cus is not to be determin'd.

When the Radiating Point a. Tab. 27. Fig. 3. T. 27. F. 3. is placed in the Focus of the Convex-Glass c d, the Rays c e, d g, after passing the Glass, run Parallel, and being produced to c x, d z, they never Intersect. Wherefore in this Case there is no Rule whereby to determine the Locus of the Object. And Barrow tells us only, Quod Remotissime Positum Aestimatur. Lect. 18. ad Finem.

(9) The Locus of an Object, beyond the Focus of a Convex-Glass, to the Eye between the Glass and Distinct Base, cannot be determined.

[Page 118] Tab. 27. Fig. 4. T. 27. F. 4. Let a be a Radiating Point placed more distant from the Glass c d than its Focus; the Rays c e, d g, after passing the Glass, do Converge towards the Distinct Base; let the Eye e f g be placed between the Glass and Distinct Base, it shall then receive the Rays c e, d g, Converging; and they being produced towards x and z separate the further.

In this and the last Section lies the great Difficulty, which the Incomparable and most profoundly Learned BARROW ( Lect. Opt. 18. Sect. 13.) confessedly passes over as insuperable, and not to be explained by whatever Theories we have yet of Vi­sion. For seeing that the Object which applies to the Eye by Less-diverging Rays, is judged the more remote; And that which applies to the Eye by Parallel Rays, is reputed most remote; it should seem reasonably to follow, that what is seen by Converging Rays, should appear yet most remote of all: And yet Experience contradicts this, and testifies, that the Point a, Tab, 27. Fig. 4. appears variously distant, according to the various Situations of the Eye between the Glass and Distinct Base; and that it does almost never (if ever) appear more distant than the Point a it self to the naked Sight, and some­times it appears much nigher: Or rather, by how much the Rays, which fall through the Glass on the Eye, do more Con­verge, by so much the nigher does the Object appear to ap­proach, insomuch, that if the Eye approach the Glass very nigh, the Object a appears in its natural Place: The Eye be­ing a little farther removed from the Glass towards the Di­stinct Base, the Point a seems yet to approach. The Eye being yet farther, the Point seems yet nigher; and so by de­grees, till at last the Eye being placed at a certain station, as at the Distinct Base, the Point a appears very nigh, so that it begins to vanish away in mere Confusion. All which (con­tinues the candid BARROW) seems repugnant, or at least not so well to agree to what we have laid down. And so he [Page 119] leaves this Difficulty to the solution of others, which I (after so great an Example) shall do likewise; but with the reso­lution of the same admirable Author, of not quitting the evi­dent Doctrine, which we have before laid down, for deter­mining the Locus Objecti, on the account of being pressed by one Difficulty, which seems inexplicable, till a more intimate Knowledg of the Visive Faculty be obtained by Mortals. In the mean time, I propose it to the consideration of the ingeni­ous, whether the Locus Apparens of an Object placed as in this oth. Section, be not as much before the Eye, as the Distinct Base is behind the Eye. Vid. Corol. 1. Prop. LVII.

(10) If an Object be more distant from a Convex-Glass than its Focus, and the Eye beyond the Distinct Base, the Locus Appa­rens of the Object is in the Distinct Base. Vid. Prop. XXXIX. Sect. 5. item Schol. Prop. L.

Tab. 27. Fig. 5. T. 27. F. 5. The Object a b c is projected by the Glass g z l in the Distinct Base d e f. The Locus of each Point in the Object is in the correspondent Point of the Image in the Distinct Base: Thus a appears at d, b at e, c at f. Gregorii Opt. Prom. Prop. XLVI.

Dechales (Dioptr. Lib. II. Prop. XI.) remarks a Matter of some moment in this Business of the Locus Objecti. The reason (says he) that the Appearances through Glasses of the Change of the Objects Place, do not so strongly strike the Sense, as the Doctrine here laid down seems to intimate, proceeds from hence; that Optick-Glasses are seldom or ne­ver made so large, as to be look'd through by both Eyes at once; for if they were, he asserts, That the Locus apparens Objecti would be much more plainly and sensibly determin'd to the sight: In this particular certainly he is much in the right; for we see at all times, that two Eyes make a more exact estimate of the Position of an Object, than one single Eye. And we have a sensible Experiment hereof in Catop­tricks, [Page 120] by the large concave Miroirs, where an Hand and Dagger presented beyond the Focus, seem to strike far with­out the Speculum, at him that presents it: But not so strongly if the Speculum be but small, so that the Image can be seen but by one Eye at once.

PROP. XXXII.

An Object being placed in the Focus of a Convex-Glass, and the Eye on t'other side the Glass, sees this Object distinctly and erect.

That an Object so posited, is seen distinctly, is evident, because the Rays from each particular Point, after passing the Glass, become parallel (by Prop. VI.) and so fall on the Eye. And therefore (by Prop. XXVIII. Sec. 7.) they are fit to cause Distinct Vision. Tab. 27. Fig. 3. T. 27. F. 3. The Point a sendeth forth its Diverging Rays a c, a d, on the Convex-Glass c d; these are transmitted by the Glass parallel in c e, d g, and so fall on the Eye; by whose Coars and Humors, especially by the Crystalline h i, they are refracted and brought together in the Point k on the Retina, there representing distinctly the Point a of the Object.

Or thus, Tab. 28. Fig. 1. T. 28. F. 1. The Eye o p, being in the place where some of the Rays from every Point in the Object are mixt together, is in the place of the greatest Confusion; and therefore by Prop. XXVIII. Sec. 3. the Vision is distinct.

I say also the Object is seen erect. In the same Figure g k is a Convex-Glass, a b c an Object; we may imagin that each Point of this Object sends from it a Cone of Rays, fall­ing [Page 122] on the whole Surface of the Glass; but for avoiding Con­fusion in the Scheme, I only express and consider the Rays c g, c h, from the Point c; and a k, a i from the Point a [...] o p is the Crystalline of the Eye. The Rays c g, c h, after passing the Glass become parallel as g o, h p, and fall so on the Crystal­line, by whose Refractions they are again brought together in the Point d on the Retina, there painting the lively Repre­sentation of the Point c; and the same may be shewn of the Representation of the Point a at f on the Retina. Where­fore the smister point c is represented on the dexter part of the Fund of the Eye d; and the dexter part of the Object a is painted on the smister part of the Eye f. And consequently the Image on the Retina being inverted, the Object is seen erect (by Prop. XXVIII. Sec. 4, 5.). Which was to be De­monstrated.

PROP. XXXIII.

An Object being placed in the Focus of a Convex Glass, and the Eye being placed in the Focus, on t'other side the Glass, sees this Object under the same Angle, as were the naked Eye placed at the station of the Glass.

Tab. 28. Fig. 2. T. 28. F. 2. a x b is an Object placed in the Focus of the Glass e f (which we suppose of the least thickness imagin­able): Let the Rays a e, b f, run parallel to the Axis x c o, these Rays are united in the Focus at o, at which Point we suppose the Eye. Produce o e and o f directly to g and k, and draw e c f (which by supposition is parallel to a x b, and then draw a c, b c. The Optick-Angle through the Glass is e o f, which we are to prove equal to a c b. Thus, a x and e c being parallel, and a e, x c, being so likewise; a x is equal to e c, and a e to x c, being the opposite Sides of a Paral­lelogram. But x c is equal to c o (the Focal Length on one [Page 123] side and t'other). Wherefore the Right-angled Triangle e c o is equal and similar to the Right-angled Triangle a x c. Where­fore the Angle e o c is equal to the Angle a c x, or e o f is equal to a c b. And consequently the Eye at o sees the Ob­ject through the Glass under the same Angle, as the naked Eye being place at c, would see it. Which was to be proved. Vid. Gregorii Opt. Prom. Prop. XLIV.

PROP. XXXIV. PROBL.

To determine the Optick Angle, or apparent Magnitude of an Ob­ject in the Focus of a Convex-Glass, to the Eye at any other station; from these Data, the Glasses Focal Length, or the Distance of the Object from the Glass x c, Tab. 29. F. 1. T. 29. F. 1. The Distance of the Eye from the Glass c d, and the Breadth of the Object a b.

The Thickness of the Glass I take no notice of, because I would not perplex the Scheme. And therefore the Glass is supposed of the least Thickness imaginable; or all the Refractions are supposed performed at the Line in the mid­dle of the Glass g e c f g.

Wherefore let a x b be an Object, and let the Rays a e, b f, fall parallel to the Axis x c o; these are united in the other Focus at o, where if the Eye be placed, it sees the Ob­ject under the Angle e o f = a c b, by the preceding Prop. Wherefore having b x the half Object, and x c; in the Right-angled Triangle b x c, we may find the Semi-optick Angle b c x. Draw the prickt Line b o directly; the Angle b o x is the natural Optick Angle of the Object b x, which is easily. obtained, in the Right-angled Triangle b o x, having b x and x o.

Let now the Eye be out of the Focus at d; the Rays a g, after passing the Glass, become parallel to e o (Prop. VI.) And therefore the Angle g d x is equal to e o x or a c x, which we have found before. Draw the prickt Line b d directly; the Angle b d x is the natural Optick Angle of the Object b x, were the Glass removed; and is easily had in the Right-angled Triangle b x d, having b x and x d (= x c + c d). So the Problem is satisfied.

PROP. XXXV. PROBL.

To determine the Visible Area of an Object in the Focus of a Convex Glass; from these Data, the distance of the Object from the Glass or the Glasses Focal Length, the distance of the Eye from the Glass, and the Breadth of the Glass.

It is shewn before in the 2d. Corollary to Prop. XXXIII. That the Visible Area of an Object in the Focus of a Con­vex Glass, to the Eye placed in t'other Focus, is equal to the Breadth of the Glass. Wherefore I shall only here consider the two other Cases.

And first, Let the Eye at o, Tab. 29. Fig. 2. T. 29. F. 2. be placed further from the Glass g l than its Focus, the Object a x b is in the Focus, g l the Glasses Breadth, x c the Glasses Focus or distance of the Object, and c o the distance of the Eye from the Glass are given. Draw g o, l o, and produce the Axis o c f infinitely. Let us now imagin the Point o a nigh Ra­diating Point or Object, sending its Rays o g, o c, o l, on the Glass. By Prop. V. Let us determin the Distinct Base or Re­spective Focus of this Point projected by this Glass, which by the foregoing Data is easily performed. Suppose this respe­ctive Focus to be c f. Draw g f, l f; where these Lines in­tersect the Object in e and d, the Visible Area of the Object is determined by e d.

Demonstration. For by Exper. 8. the Progress of a Ray through different Mediums is Reciprocal. Wherefore if the Rays o g, o l, be refracted into g e, l d; it will necessarily fol­low, that e g, d l, being consider'd as two Rays flowing from the Points e and d, they shall be refracted into g o, l o; where­fore this Analogy will hold, f c : c l : : fx (= f c − xc) : x d. Which x d is half the Visible Area to the Eye at o.

[Page 127] Secondly, Tab. 29. F. 3. T. 29. F. 3. Let the Eye at o be placed nigher the Glass than its Focus. Draw g o, l o. And produce the Axis o c infinitely backwards towards f. Then (as before) supposing the Point o a Radiating Point or Object; by the Data, and by Prop. VIII. we may easily determin the Ima­ginary Focus thereof, which let be c f. From f draw f g di­rectly to e, and f l to d. e d, by the foregoing Demonstra­tion, is the Visible Area of the Object a b. And c f : c l :: f x (= f c + c x) : x d. Which is half the Visible Area to the Eye at o.

PROP. XXXVI.

An Object being placed more distant from a Convex Glass than its Focus, and the Eye placed on t'other side this Glass nigh­er to the Glass than the Distinct Base to the Glass, sees this Object erect and confused

I have shewn before ( Prop. V. XXVI, XXVII.) that if an Object be placed more distant from a Glass than its Focus, this Object is projected in a distinct Base somewhere on t'other side the Glass. And that this Distinct Base, may be some­times Less, sometimes equal to, and sometimes Greater than the Object it self.

[Page 128] Tab. 30. Fig. 1. T. 30. F. 1. a b c is an Object exposed to the Glass g l, and projected in the Distinct Base f e d. Each Cone of Rays from the Object is made to converge, as b l g is formed into g e l. Let the Eye be placed at k. I say, first the Ob­ject appears erect. For the Rays from a the upper part of the Object tend to, and determin in the lower part of the Fund of the Eye. And the Rays from c the lower part of the Object tend towards the upper part of the Retina; and there­fore the Appearance is erect, (by Prop. XXVIII. Sec. 4, 5.)

Moreover, the Eye being placed any where between the Glass g l, and the Distinct Base f e d, does not receive the Spe­cies of the Object inverted; for that only happens just in the Distinct Base it self. And on this account, 'tis manifest, that the Appearance to the Eye in this posture shall be Erect.

I say likewise, that the Apearance of the Object is confused. Let us conceive the Pupil of the Eye at m n; here it receives the Rays from the Point b converging; and therefore by Prop. XXVIII. the Representation of the Point b on the Fund of the Eye shall be confused. And so of the other Points in the Object, that is, of the whole Object. And if the Eye recede from the Glass, as to h i, the Vision is more confused, because the Rays are nigher converged.

PROP. XXXVII. PROBL.

To determin the Optick Angle, or Apparent Magnitude, and the Visible Area of an Object placed as in the last; from these Data, The Breadth of the Object a b c (Tab. 30. Fig. 2.), T. 30. F. 2. Its Distance from the Glass z b, The Glasses Power, The Glasses Breadth g l, And the Distance of the Eye from the Glass k z. [Page]

[Page] [...]

[Page 129] (1.) From these Data by the foregoing Doctrine of Con­vex Glasses, 'tis easie to determine the Breadth of the Distinct Base. For by Prop. XXVI, XXVII. As the Distance of the Object from a Convex Glass: To the Breadth of the Object :: So the Distance of the Image in the Distinct Base : (which is easily obtained from the foregoing Data) To the Breadth of the said Image. That is (in the foresaid Figure) b z : a c :: e z : d f. And As a b : b z :: d e : e z. And consequently the An­gle e z d is equal to the Angle a z b; the Triangles e z d and a z b being similar. And in the Right-angled Triangle a b z, we have a b and b z, to find the Angle a z b = e z d; which is the Angle, under which the Object would appear to the Eye placed just touching the Glass at z; supposing the Glass to be there of the least Thickness imaginable.

(2.) Let us now suppose the Eye at k. I say if the Rays from the single Points a, c, in an Object be converged by the Convex Glass g l to any other Points d, f; the Object shall appear under the Optick Angle x k m, as do the Apices of the Pencils from the Extreme Points of the Object, which is d k f = x k m. Vid. Gregorii Opt. Promot. Prop. X+LV.

(3.) Wherefore in the Right-angled Triangle e k d, we have e k and e d, To find the Angle e k d = z k x, which is the Semi-Optick Angle to the Eye at k. In like man [...]e [...] may we find the Optick Angle at any other given Station of the Eye, as at n, e n d = g n z being equal to half f n d.

(4.) As to the Visible Area of the Object, 'tis determin'd as before, Prop. XXXV. and Schol. I shall only add, That if the Eye be at q. and the Line d q being drawn, and produced towards h, it does not fall upon the Glass g l▪ then the Eye at q shall not see through the Glass the Point a, the Apex of whose Pencil is in the Point d. And from hence we may per­ceive, that at k, we may see through the Glass a greater space than the Area of the Object a b c. At [...] we can but ju [...] see [Page 130] the whole Object a b c; and at q we cannot see extreme Point a c. By which the 81 and 82 Prop. of Kepler's Diop­trick [...] are manifest.

(5.) If the Glass have any Thickness considerable, see Prop. XXXI. concerning the Locus Objecti, Sec. 7.

PROP. XXXVIII.

An Object being more distant from a Glass than its Focus, and the Eye placed in the Distinct Base, the Object appears most con­fused.

By Prop. XXXVI. 'tis manifest, that the nigher the Eye approaches the Distinct Base, the more confused the Object appears; because the Rays from each particular Point, do fall on the Eye closer Converged, and more orderly separated by the Glass from the Rays of adjacent Points. But in the Distinct Base, the Rays are most of all Converged, and most orderly united in their proper Points, each with the Rays flowing from the same Point in the Object. Wherefore the Eye is there in the greatest Confusion.

Moreover (by Prop. XXVIII. Sec. 3.) 'tis requisite to Di­stinct Vision, that the Rays from the several Points in the Ob­ject, fall on the Eye altogether confused thereon (as they do on the Glass in the Hole of a dark Room), that the Coats and Humors of the Eye may refract them, and bring them toge­ther, [Page 131] distinctly painting the Image on the Retina. But when the Eye is in the Distinct Base, all this is frustrated.

PROP. XXXIX.

The Object being more Distant from the Glass than the Focus; and the Eye further from the Glass than the Distinct Base, begins to perceive the Object inverted; and, at a proper di­stance, Distinct.

(1.) This is manifest by Tab. 30. Fig. 3. T. 30. F. 3. wherein a b c is an Object, g s l the Glass projecting the Distinct Base f e d, o is the Eye. And because half the Pencils of Rays that flow from each of the extreme Points of the Object a and c, pro­ceeding forwards from the Points d and f in the Distinct Base, after they cross in the aforesaid Points d and f, by reason of their too great Divergence, do escape the Eye. Therefore, for avoiding Perplexity in the Scheme, I have only expressed the Half Cones s a l, s d l, and s c g, s f g, which incurr the Eye.

(2.) Wherefore we may observe, that the Diverging Rays p f o and o d q, falling diverging and confused on the Crystalline p q, are thereby collected, and made to concurr on the Retina in the Points n and r; and there paint the upper Point a of the Object on the upper part r of the Eyes Fund; and the lower Point c of the Object on the lower part n of the Retina. Therefore by Prop. XXVIII. Sect. 4, and 5. the Object appears inverted, because the Image represented on the Retina is Erect.

(3.) Or more plainly thus, We may now conceive the Di­stinct Base it self f d to be as it were an inverted Object (as here the Cross turned with its head downwards). Therefore the appearance thereof to the Eye, shall seem inverted. For if the Eye perceive an erect Object, erect, it must needs per­ceive the Distinct Base, being an inverted Object, inverted.

[Page 132] And this may be laid down as a General Rule, That where an Object, or the Image, or Representation of an Ob­ject, which is next the Eye, is inverted, there the Object shall appear inverted. And where the Object, or Image next before the Eye, is erect, the Object appears erect.

(4.) I say also, At a proper distance the Inverted Appearance is Distinct. For when the Eye is so far removed from the Di­stinct Base, as to be able to correct the Divergence of the Rays, that come to it from each Point in the Distinct Base, and to form them into correspondent Cones determining their Apices on the Retina, the Vision is Distinct, otherwise not.

Moreover there are two other Phaenomena of this Posture of Object, Glass, and Eye; which I shall here explain as fol­lows.

(5.) If the Eye be moved upwards towards z, the Object shall appear to move downwards. If the Eye be moved downwards towards x, the Object seems to move upwards. To explain this, let us consider the Point c; and how the appearance thereof is brought to and formed in the Eye. And we shall find, that in the Posture of the Scheme, the Point c is expressed in the Eye, only by that parcel of its Rays that fall on the half Glass s g, for these are brought together in s f g, and flowing forward, be­come p f o: But the other parcel of Rays, s c l, that fall on the half Glass s l, and are brought together in s f l, proceeding forwards, escape the Eye; for they go on in p f z. Where­fore the Point c is now represented through the half Glass s g, or seen somewhere between g and s. But the Eye moving up­wards towards z, so that it miss the Rays p f o, and receive only the Rays p f z; the Point c shall then be represented through the half Glass s l, or seen somewhere between s and l, and consequently shall seen to move downwards. What is shewn of the Point c, may be conceived of the other Points in the Object, that is, of the whole Object.

[Page 133] And this Phaenomenon is the strongest Confirmation imagina­ble, that the Locus Apparens Objecti, in this Case, is in the Distinct Base (as is said before, Prop. XXXI. Sec. 10.). For by raising the Eye, we depress the Point f; and by depressing the Eye, we raise the Point f. Wherefore. the appearance of the Point c is at f. For the Point c in the Object seems to move contrary to the motion of the Eye.

From the foregoing Explication it follows, that if the Glass be moved upwards, the Object appears to move upwards. And if the Glass be moved downwards, the Object appears to move downwards; for the moving of the Glass upwards is the same as moving the Eye downwards; but moving the Eye downwards, makes the Object appear to move upwards; therefore the moving of the Glass upwards makes the Objects appear to move upwards.

Or thus, Because moving the Glass upwards, the Distinct Base is moved upwards (for it follows the Glass), and the Di­stinct Base is in this case as it were the Object. ( Prop. XXXI. Sec. 10.) Therefore by the Glasses motion upwards, the Eye shall perceive the Object moved upwards.

(6.) If with both Eye we look at the inverted Appear­ance of an Object through a Convex-Glass, we shall perceive the Object double. Supposing the Eyes not very far remov­ed from the Distinct Base. Tab. 30. Fig. 4. T. 30. F. 4. a b c is an Ob­ject projected by the Glass g l in the Distinct Base f e d; the Point e being correspondent to the Point b. Let the Cone of Rays g e l flow forwards from e, and become z e y. Let the two Eyes n, m meet the Cone of Rays z e y. I say the Eye n perceives the Point e (the Representation of the Point b) by the Rays z e q, which flow forward from i e l; and consequently the Eye n perceives the Point [...] or e, as it were, somewhere between [...] and l. In like manner, the Eye [...] perceives the Point b or e somewhere between s and g: Wherefore the two Eyes seeing the Point [...]. or e, as it were, in two places, see it [Page 134] Double. What is said of the Point b, may be understood of any other Point in the Object, and consequently of the whole Object. And if we shut the Right Eye m, the Left Appear­ance of b vanishes; if the Left Eye n, then the Right Appear­ance of the Object vanishs.

But if the Eyes be removed very far from the Distinct Base f e d, their Interval continues the same; and the Rays e k, e p, Diverging continually, they shall at last fall on the Eyes; that is, e k on the Eye n, and e p on the Eye m. By which the Eye n shall perceive e as it were at u (k e u being one Right Line), and the Eye m shall perceive the same e as it were at r (p e r being one Right Line), whereby the Difference of the Places r and u becomes less than before, till at last the Eye be got at such a Distance from the Distinct Base, that this Difference of the Apparent Places becomes so small, that 'tis insensible to the sight.

Quaere. How Dr. Briggs would explain this Phaenomenon of Double Vision by his Theory of sight. Philosoph. Collect. Numb. 6.

PROP. XL.

To determine the Optick-Angle or Apparent Magnitude, and the Visible Area of an Object, placed as in the last, from these Data; The Breadth of the Object a b c, (Tab. 30. F. 5.) T. 30. F. 5. Its Distance from the Glass s b, The Glasses Power or Focal Length, The Glasses Breadth g l, And the Distance of the Eye from the Glass o s.

(1.) It is shewn before ( Prop. XXXVII.) how we may from these Data obtain the Breadth of the Distinct Base f e d, and its Distance from the Glass e s. Wherefore o s − e s = o e. Then in the Right-angled Triangle o e f, we have o e and e f to find the Angle f o e, which is the Semi-Optick Angle, un­der [Page 135] which the Eye perceives the Inverted Appearance of the Object. In like manner may we find the Optick Angle to any other station as at n.

(2.) The Learned GREGORY in his Opt. Promot. Prop. 46. has a curious Theorem to this purpose, 'tis this, The Rays from each single Point in an Object a b c, being formed by the Glass g s l into the Distinct Base f e d; and the Eye o being placed behind the Points of Concourse f e d; the Image of each Point in the Object shall ap­pear in the Apex of its Pencil ( That is, The Image of the Point a shall appear in the Apex of its Pencil at d, &c.), And the Object shall appear Inverted; And under that Optick-Angle, as the Apices of the Pencils of the Extreme Points of the Object; That is, under the Angle f o d.

(3.) The Visible Area is thus determined, Tab. 30. Fig. 6. T 30. F. 6. The Object a b c is projected by the Glass g s l in the Distinct Base f e d. Let the Eye be at o; draw o g, o l; the Eye at o shall see no more of the Object, than what is represented between z and r, where the Lines o g, o l, intersect the Di­stinct Base. To determine the Points in the Object, that an­swer the Points z, r, in the Distinct Base. From the Centre of the Glass s (for I suppose it of no Thickness), draw z s x, r s k; the Point k answers r, and x answers z. Lastly, from the foregoing Data to determine the Measure of z r say, As s o : s g :: e o: e z = ½ z r; but the three first are known, there­fore the fourth is known also. Which was to be found.

(4.) If the Line o y produced do not fall on the Glass, then that Point in the Object, whose Apex of its Pencil is at y, or which is projected in the Distinct Base at y, shall not be seen through the Glass by the Eye at o. Vid. Schol Prop. XLVI. Gregorii Opt. Promot.

PROP. XLI.

An Object being placed nigher a Convex Glass than its Focus, and the Eye on t'other side the Glass, at any Distance within the Eyes power, sees this Object Distinct and Erect.

I say, Within the Eye's Power; for 'tis shewn before in Prop. VII. and VIII. That when an Object is placed nigher a Con­vex Glass than its Focus, the Rays from each Point thereof, after passing the Glass, do Diverge, but not so much as if the Glass were away. Wherefore, tho the Eye looking at an Object within the Focus of this Glass, may be nigher the Ob­ject, than if the Glass were away, and yet see it distinctly; yet there is a mean to be observed; for even with the Glass it self the Eye may be too nigh the Object, and not able to correct the Divergence of the Rays it receives.

But that an Object thus posited appears Distinct to the Eye at a convenient Distance, is manifest; for the Rays from each Point proceed moderately Diverging, and so fall on the Eye. Prop. XXVIII. Sec. 7.

That an Object thus posited is seen Erect, is evident from Prop. XXXII. and XXXIX. [Page]

[Page] [...]

PROP. XLII.

To determine the Visible Area, and the Optick-Angle or Apparent Magnitude of an Object placed as in the last, from these Data, The Power or Focal Length of the Glass, The Distance of the Object from the Glass, The Distance of the Eye from the Glass, and the Breadth of the Glass.

Tab. 31. Fig. 1. and 2. Tab. 32. Fig. 1, 2. a b is an Object exposed to the Glass g l, nigher to it than its Focus. o is the Eye, which in F i g. 1. is supposed nigher the Glass than its Focus; and this is the First Case. But in Fig. 2. the Eye o is supposed further from the Glass than its Focus; and this is the Second Case. 'Tis shewn before ( Prop. V. VIII.), how the Respective and Imaginary Foci f are determined in one and t'other Case.

First let the given Breadth of the Glass be g l. By Prop. XXXV. we determine the Visible Area z y. Let us then suppose the given Breadth of the Glass to be p q. By the same Prop. XXXV. we determine the Visible Area e d.

As to the Optick-Angles g o l (supposing the Object z y) or p o q (supposing the Object e d) they are determined from the Data, by Plain Trigonometry of Right-angled Triangles, as before in Prop. XXXIV.

And we shall find by Calculation (as indeed 'tis evident by the very Inspection and Consideration of the Schemes), that the Collateral Parts z e, d y, of the Object z y, are much more magnifid (in respect of their Natural Appearance) by Broad Glasses formed on small Spheres, than the Middle Parts e x, d x; for the Angle g o p is the Optick-Angle, through the Glass, of the Part z e; and the Angle p o s is the Optick-An­gle, through the Glass, of the Part e x; but the former ex­ceeds the Natural Optick-Angle much more than the latter. [Page 138] As to the Natural Optick-Angle of the Object z x, were the Glass away; draw z o directly ( Fig. 1.), then in the Right-angled Triangle z x o, we have z x and x o to find the An­gle z o x, which is the Natural Optick-Angle of the Object z x.

From hence it is, that by Broad Glasses formed on small Spheres, the Extreme Parts of strait Objects, seem to be in­curved and bent; as is manifest in the Case of the Micrometer, or Lattic [...]e of fine Hairs, strained before the Eye Glass in a Telescope, for Measuring the Diameters of Objects. As Pere Cherubin complains in his Dioptrique Oculair. Part. III. Sec. 7. Chap. 1. pag. 239. but understood not the reason. Of this we may make Experiment, by looking with a very Convex Glass at two Parallel Lines drawn pretty close on a Paper.

PROP. XLIII.

All Objects seen through Concave Glasses appear Erect and Di­minish'd.

That the Object shall appear Erect is manifest; for 'tis before ( Prop. XXXIX. Sec. 3.) laid down as a General Rule, That where the Object, or Image that is next before the Eye, is Erect, the Eye shall perceive the Object Erect; and where Inverted, the Eye sees it Inverted: But Concaves have no Distinct Base (as Convexes have) wherein they represent the Image of the Object Inverted, and consequently, wherever the Eye is placed behind a Concave Glass, it shall perceive the Object through it, in its Natural Posture. That Concaves have no Real Di­stinct Base, is most plain from the Doctrine that foregoes re­lating to them: For a Distinct Base is caused by the Col­lection [Page]

[Page] [...] [Page 143] of the Rays proceeding from a single Point in the Ob­ject, into a single Point in the Representation; but Concave-Glasses do not unite, but scatter and dissipate the Rays. 'Tis true indeed, a Convex-Glass may cause a Distinct Base, not­withstanding a Concave placed behind it or before it (as we see in Prop. XV. XVII, XVIII.); but then this Distinct Base pro­ceeds not from the Concave, but from the Convex: For the Concave exerts its scattering power even in this Case; and it protracts the Uniting of the Rays to a greater Distance from the Convex, as in manifest from the fore-cited Propositions.

I say also, The Appearance of Objects through Concaves is dimi­nish [...]d. Tab. 32. Fig. 1. T 32 F. 1. a x b is an Object, o the Eye. Draw a e o, b d o, directly strait from the Extremities of the Object to the Eye. The Angle comprised by the direct Rays a e o, b d o, that is, the Angle a o b is the Natural Optick-Angle. Let now the Concave-Glass c f be interposed; here, because the Rays a e, b d, would naturally concurr at o, now the Con­cave-Glass is interposed, their Concourse shall be protracted beyond o; wherefore the Eye at o shall not perceive the Ex­tremities a, b, of the Object through the Glass by the Rays a e, b d, but by some other Rays, and these other Rays must either fall without (that is farther from the Axis o x, than) a e, b d, or they must fall within a e, b d: But they cannot fall with­out a e, b d; for if a e, b d, themselves be made by the Glass to concurr beyond the Eye at o, much more shall the Con­course of any other Rays that fall without a e, b d, be protra­cted beyond o; and consequently they cannot convey the Ap­pearance of the Extreme Points a, b, to the Eye at o. Where­fore it remains, that the Rays that do this, must fall within ae, bd. Let us suppose these Rays to be a c, b f, which in their Natural direct Course, would concur at q, but by the Refractive Power of the Concave, are bent and made to pro­ceed in c o, f o, their Concourse being protracted till they arrive [Page 144] at the Eye in o. Here the Optick-Angle through the Glass is c o f, which is less than the Natural Optick-Angle a o b, and consequently the Appearance of the Object is diminish'd. Which was to be proved.

PROP. XLIV.

Concerning the Distinct and Confused Appearance of Objects through Concaves; as also of their Faint or Obscure Ap­pearance.

We are here to remember what before is laid down, concern­ing the Virtual Focus of a Concave exposed to Parallel or to Diverging Rays (in Prop. XI, XII, XIII. & Corol. Prop. XV.). We are likewise to take notice, that in Plain Vision, when the Rays from any single Point in an Object do not Diverge more that what the Refractive Power of the Coats and Humors of the Eye can correct; so that these Rays may be brought to­gether in a Correspondent Point on the Retina; then the Ap­pearance of that Point is Distinct. For instance, Tab. 32, F. 2. T. 32. F. 2. Let the Point a diffuse the Rays a b, a s, a r, a c. Suppose the Breadth of the Pupil were p q, and the Eye there placed; perhaps the Refractive Power of the eye is not sufficient to correct the Divergence of the Rays a p b, a q c. But if the Pupil (continuing of the same Breadth) recede to r s, then only the Rays a s, a r, fall into it; and these perhaps may be reduced by the Eye to determine in a Point on the Retina; because they do not Diverge so much as the former. If therefore we suppose the Pupil in its former station at p q, but now only to be of the Breadth i d, so as to admit only the Rays a d s, a i r, then the Point a may be seen as distinctly as at r s. This manifest even by Experiment; for apply a Minute Object so near the Eye, that it appears very confus­ed; then place before the Eye a very small Hole made with [Page 145] a Pins end in a Paper; The Object shall now appear Distinct. For the Hole in the Paper serves to make the Pupil more Nar­row. Which evidently proves what we have laid down.

The same may be shewen concerning Vision through a Con­cave Glass Tab. 32. f. 3. T. 32. F. 3. a is a Radiating Point, whose Rays a b, a c, fall on the Concave b c. These after passing the Glass Diverge more than before. Let p q be the Breadth of the Pupil, receiving the refracted Rays a p, a q, These Diverge so much, that 'tis not in the power of the Eye to reduce them, and form them into an inward Cone, determining its Apex on the Fund of the Eye. But let the Pupil continue of the same Breadth, and recede to r s, where it may only receive the Rays a d s, a i r; and then perhaps the Eye may prevail to reduce these; be­cause they do not Diverge so much as the other Rays that fell on the Pupil at p q. Or otherwise, let the Pupil continue at p q, But let its Breadth be contracted to d i; so that it may receive no more Rays in this Posture, than (continuing of its natural Breadth) at r s. Here likewise the Appearance of the Point a may be Distinct. Of this likewise we may make a most con­vincing Experiment: For take a Concave of a small Sphere, and place it very near the Eye, and the Appearance of distant Objects through it is Confus [...]d. Remove it farther from the Eye, and the Appearance shall be more Distinct. But even in a Posture where the Appearance is Confused, contract the Pupil by placing before it a small Hole Prick [...]d in a Paper, And the Object shall appear more Distinct, though Obscure, which upon the removal of Hole, shall be again Confused, though more Lightsome. Vid. Kepleri Dioptr. Prop. C.

Wherefore if we have the Distance of a Radiating Point from a Concave, and the virtual Focus of the Concave, and the Di­stance of the Eye behind the Glass; we may easily by the fore­going Rules ( viz. Corol. Prop. XV.) find the virtual Focus of these Rays. As suppose in the same Fig [...] the Rays flowing [Page 146] from a, after passing the Glass, Diverge as if they came direct­ly from z. And consequently, if in Plain Vision the Eye at r s be able to see distinctly the Point a, if it were removed nigher, as at z, then it shall be able to see the Point a distinctly through the Glass. But whether any Particular Eye be able to do this, is impossible to be known by Rule; The Strength and Weakness of Mens Eyes being infinitely various. And therefore one Man may see a Point at a certain Distance distinctly through a Glass, which Glass to another Man would render it Confused; As 'tis Plain in the Case of Myopes, or Short-sighted Persons.

As to the Strong or Faint Appearance of Objects through Concaves; Because the Concave ( Tab. 32. f. 3.) T. 32. F. 3. scatters the Rays flowing from the Point a, insomuch that now the Rays a b, a c, scape the Pupil p q, the Point a shall appear more Faint through the Glass than naturally. For Strong or very Lu­minous Virsion proceeds from a greater Quantity of Rays or Light entring the Pupil; And Faint or Obsure Vision from a less Quan­tity of Rays.

PROB. XLV.

Concerning the Apparent Place of Objects seen through Concave Glasses.

Tab. 32. f. 4. T. 32. F. 4. If the Point a Radiate on the Concave Glass c d, and the Rays after passing the Glass Diverge as if they came directly from the Point b. And the Eye e g receive all or part of the Rays, and there by perceive the Radiating Point a, the Point b is the Locus Apparens of the Point a. That is, the Point a is seen by the Eye, as if it were at b.

By which we may Observe, that the Apparent Place of Ob­jects seen through Concaves is brought nigher the Eye. And hence 'tis manifest, why they help their Eyes, who are short-sighted, or can only see nigh Objects. For these Glasses make distant Objects seem nigh. Vid. Dechales Dioptr. Lib. 2. Prop. XXXIX.

Here also by the way we shall Note, That suppose a Pur­blind Person, that can Read distinctly or see Objects at the di­stance of a foot from his naked Eye: A Concave Glass, whose virtual Focus is a Foot distant from it, makes such a Person see distant Objects distinctly. Wherefore knowing the Distance at which a Purblind Person Reads distinctly, 'tis easie to assign him a proper Glass for his Eye, to see distant Objects. Vid. second Part. C. 3.

Tab. 32. f. 5. T. 32. F. 5. Is in all things Correspondent to the Doctrine laid down in Prop. XXXI. Sec. 7. and marked with the same Let­ters; so that the very Words of that Section may be applied to the Concave, as well as to the Convex. Only the Imaginary Focus e for the Concave is to be determin'd by Prop. XI, XII, XIII. XV. and Corol. Prop. XV. &c. I shall not therefore, Repeat, but refer to that Section.

PROP. XLVI. PROBL.

To determine the visible Area, and the Optick Angle or Apparent Magnitude of an Object seen through a Concave from these Data; The Power or Focal Length of the Glass; The Distance of the Ob­ject from the Glass; The Distance of the Eye from the Glass, and the Breadth of the Glass.

Tab. 33. f. 1. T. 33. F. 1. g z l is a Concave Glass, whole virtual Focus is given. Likewise g z the half Breadth of the Glass is given also. a b c is an Object, whose Distance from the Glass b z is given. And the Distance of the Eye o from the Glass, o z, is also given.

Let us now conceive the Point o an Object or nigh Radiating Point, sending its Rays o g, o z, o l, on the Concave Glass; These (by t [...]e foregoing Doctrine of Concaves) after passing the Glass Diverge more than before their Entrance, and proceed in g a, l c, as if they came directly from a certain Point f. This Point f or the Line f z, is easily determind by Prop. XI, XII, XIII. XV. and Corol. thereof. Wherefore having f z, we may say, As f z : z g :: f b (= f z + z b) b a. Which is half the Visible Area.

As to the Optick Angle through the Glass, g o z, 'tis easily obtain [...]d in the right-angled Triangle g o z, having g z and z o.

Draw a o directly, The Angle a o b is the natural Optick Angle, under which the Object would appear to the Eye, were the Glass removed; This we obtain in the right-angled Triangle a b o by having a b and b o = b z + z o.

PROP. XLVII.

The further the Eye is removed from the Concave Glass, the Ob­ject appears the less. Zahn Telescop: Fund. 2. Synt. 2. Cap. 5. Prop. XXIII. Dechales Dioptr. Lib. 2. Prop. LII.

This is manifest from Tab. 32. f. 5. T. 32. F. 5. For we are to Consider the Image d e f, in the virtual Focus, as the Object, and as look'd at without the Glass. For the Lines d o, f o, which de­termine the Optick Angle, are drawn directly to the Eye at o. Wherefore if we conceive the Eye o removed farther from the Glass; the Angle d o f must needs decrease, and consequently the Object appear less. What is here said of the Concave may be ap­ply'd to the Convex in Tab. 31. f. 3. [...]. 31. F. 3.

PROP. XLVIII.

If a Concave Glass be removed from the Eye, so large an Area or space of the Object cannot be seen through it.

This is the 97 th Prop. of Kepler's Dioptricks. Wherein he seems to make no Distinction between removing the Glass from the Eye, and removing the Eye from the Glass. Whereas there is a very great Difference between both Motions to be consider'd in Dioptricks. For in moving the Glass from the Eye, the Glass Approaches the Object, and the Locus Apparens Objecti is chang­ed. But in moving the Eye to or from the Glass, the Locus Appa­rens Objecti never alters. Prop. XXXI. Sec. 7. And from his Proof of this Proposition 'tis manifest, that he should have ex­pressed it thus,

If the Eye be removed from a Concave, so large a space of the Object cannot be seen through it.

Wherefore instead of the foregoing 48 th. Proposition, I sub­stitute this; Which is Evident by our foregoing Method from Tab. 33. f. 3. T. 33. F. 3. Wherein a b c is an Object, g l a Concave Glass, o the Eye nigh the Glass, e Eye more Distant from the Glass. Let the Point of Divergence, answering to the Station at o, be f (determin'd by the foregoing Doctrine): draw f g a, f l c, directly; the space visible at o is a c. Now, when the Eye is removed from the Glass to e, the Point of Divergence shall also be removed from the Glass, and determin'd by what fore­goes, as suppose at q (till the Eye be at an infinite Distance, and then the Point of Divergence is as far from the Glass as 'tis possible, viz. in the virtual Focus): draw q g n, q l m, directly; The space visible at e is n m less than a c. Which was to be Demonstrated.

[Page 152] Or otherwise. Tab. 34. f. 1. T 43. F. 1. d e f is the Locus Apparens Ob­jecti abc through the Glass g l; the Eye is at o; o g, o l produced directly meet the Image in the Points d, f. Wherefore exactly the whole Object a b c, and no more, is seen through the Glass g l by the Eye at o. Let the Eye be removed to q; If we draw q d, q f, these fall not on the Glass, and consequently the Extremities of the Object a and c shall not be perceived by the Eye at q through the Glass. ( Schol. 1. Prop. XLVI.) Draw therefore q g, q l, and produce them directly to n and m; the Points in the Object y x, answerable to the Points n, m, are the utmost Extremities visible through the Glass g l to the Eye at q. Which Point y, x, are e [...]sily determinable by what is laid down Prop. XL. Sec. 3. For from the Center of the Glass z, draw z n y, z m x, directly; y, x, are hereby determin'd.

As to this 48 th. Prop. VIZ. That if a Concave Glass be removed from the Eye, so large an Area or space of the Object cannot be seen through it. 'Tis needless to enlarge thereon, after our 46 th. Prop..

PROP. XLIX.

The farther a Concave Glass is removed from the Eye, The Ob­j [...]s are thereby the more diminish [...]d, as long as the Glass continu [...]s nigher to the Eye than to the Object,

This is the 98 th. Prop. of Kepler's Dioptricks. We may find it also, In Cherubin's La Dioptrique Oculaire pag. 77. Dechales Dioptrica Lib. 2. Prop. XXXVIII. Herigonii Dioptr. Prop. XXXI. But in all of them obscurely and loosely proved.

The Proposition is universally True, as well in Convex as Concave Glasses, only with this Restriction in the Convex, That the Object is to be nigher the Glass, than the Glasses Focal Length. And then we may express the Proposition univer­sally thus, [Page] [Page 152] [...] [Page]

[Page] [...]

[Page 153] A Convex-Glass being equally Distant from the Eye and from the Object, renders the Appearance the most Magnified; and a Concave the most Diminished, that That Distance of Eye and Object will allow.

For the Proof hereof I need offer no more than the follow­ing Calculations (instituted according to the Doctrine Prece­dent) wherein we shall find ( Tab. 31. f. 3. and Tab. 32. f. 5.) T. 31. F. 5. T. 32. F. 5. the Angle d o e, (which is the Angle under which the Object appears through the Glass) in the second Case (wherein the Glass is equally Distant from the Eye and Object) to be great­er for the Convex, and less for the Concave, than the same Angle in either of the other Cases.

And by the same Calculations we may likewise observe in the first and third Cases, That when the Glass is equally remo­ved from the middle towards the Eye, or towards the Object, the Appearance is equally magnified by the Convex, and dimi­nish'd by the Concave; for we find in the first and third Cases, the Angles d o e equal.

I say, this may be sufficient for the Proof of the foregoing Position. But to this I shall add also this farther Considerati­on. That by the preceding Doctrine, the magnify'd Appear­ances of Objects through Convexes, and their diminish'd Ap­pearances through Concaves, being deduced from the Loci Ap­parentes of Objects through those Glasses. We may easily con­ceive, That supposing the Convex-Glass g l (Tab. 34. f. 2. T. 34. F. 2.) or Concave g l (Tab. 34. f. 3. T. 34 F. 3.) to touch the Eye o, and that the Apparent Place of the Object a b c is d e f. The Eye perceives the Object under its own Natural Optick-Angle, neither mag­nify'd or diminish'd by one or t other Glass.

Likewise if, in the same Figures, we conceive the Glasses (which are still supposed of the least thickness imaginable) to touch the Objects, the Object and Locus Apparens thereof are the same; and consequently, the Optick-Angle in this Posture can neither be increased or diminish'd by either of the Glasses.

[Page 154] Wherefore it remains, that seeing the Glasses do not at all Exert their Effects in either of the Extremes, that is, either touch­ing the Eye, or touching the Object: And seeing they Exert their Effects equally being equally removed from the middle z between the Eye and the Object; It will follow, that they Exert their Effects most powerfully being placed just in the middle z between the Eye and Object, that is, the Convex by Magnifying, and the Concave by Diminishing the Appearance.

And thus much shall suffice concerning single Glasses apply'd to the Eye.

[Page 155]

Hitherto we have Treated of single Convexes and Concaves, apply'd to the Eye. I proceed now to the Combination of Con­vexes with Convexes, and Convexes with Concaves; Wherein the Properties, Effects, and Appearances of Telescopes, and Mi­croscopes, of all kinds shall be declared.

PROP. L.

The Telescope, consisting of a Convex Object-Glass and a Convex Eye-Glass of a less Sphere or greater Convexity, is explained.

Tab. 35. f. 1. T. 35. F. 1. Let there be a Distant Object such as A B C; From whose highest Point A let the Rays a a a, mark'd by the long Pricks, proceed. And from the middle Point B, let the Rays expressed by the continued Lines b b b, proceed. And from its lower Point C, the Rays mark'd by the round Pricks c c c. And so Rays from all the other Points in the Object. These falling on the Object-Glass x y z, are formed thereby into the Distinct Base f e d. Let now the Eye-Glass g h l be placed as far distant from this Distinct Base f e d, as is the Focus of this Eye-Glass, that is; Let e h be the Focal length of the Eye-Glass g h l (And consequently the Distance of the Glasses y h is the Aggregate of both their Focal lengths) And let the Eye o be placed as far distant (or rather a little more distant) from the Eye-Glass g h l as is the Focal length of the same Eye-Glass. I say the Eye shall perceive the Object ABC, Distinct, Magnified, and Inverted.

First, I say the Object is seen distinctly. For the Rays from each Point, being made by the Object-Glass to Converge to­wards the Distinct Base, proceed forward from the Distinct Base Diverging, and so fall on the Eye-Glass. Thus the [Page] [Page]

[Page]

[Page 157] Rays b x, b y, b z, from the middle Point of the Object B▪ are made to Converge into x e, y e, z e; And crossing at e, they flow forward, and fall on the Eye-Glass about h Diverging. Wherefore the Point e being now in the Focus of the Eye-Glass, the Rays that flow from it upon the Eye-Glass, after passing the Eye-Glass become parallel ( Prop. VI.) and fall so on the Eye at o; By whose Coats and Humors they are re­fracted and brought together on the Point s on the Retina (Prop. XXVIII. Sec. 7.) there painting the lively Image of the Point B in the Object.

Secondly, In like manner may we conceive the Representation of the Collateral Points: thus the Rays a x, a y, a z, proceed­ing from the upper Point A of the Object, are by the Object-Glass x y z made to Converge towards the Distinct Base in d; From whence flowing forward they fall Diverging on the Eye- Glass at l; by which they are made to run parallel amongst themselves, and are bent towards the Focus at o; where falling on the Pupil parallel, they are by the Eye refracted and brought together in the Point r on the Retina. There paint­ting the Representing of the Point A in the Object.

Thirdly, That the Rays flowing Diverging from each parti­cular Point in the Distinct Base f e d are brought by the Eye-Glass to a Parallelism amongst themselves, is manifest from Prop. VI. And that the Rays from f, from e, and from d, are mixt and confused by the Eye-Glass in its Focus at o (or thereabouts) is manifest from hence, that we may conceive one Ray in the Cone of Rays g f, or in the Cone of Rays z f x, that runs parallel to the Axis y e h o s, or at least that would run so parallel, if the Breadth of the Object-Glass in respect of the Breadth of the Eye-Glass will permit. This Ray, I say, by the known Properties of the Convex Eye-Glass, shall be refracted into its Focus at o; and all the other Rays of the same Cone f g, after passing the Eye-Glass, shall [Page 158] be refracted and made to proceed parallel to that single Ray, that is bent into the Focus o, because the Point f is supposed in t'other Focus of the Eye-Glass. Wherefore the Rays from all the Points in the Distinct Base are confounded together about the Focus of the Eye-Glass.

Fourthly, Or otherwise I shall explain this matter thus; If we conceive y, a Radiating Point, sending forth the Rays y g, y b, y l : And the difference between the Focal length of the Object-Glass and Focal length of the Eye-Glass to be very great (as suppose the Object-Glass to be twelve Foot, and the Eye- Glass three Inches) we shall find by Prop. V. the Point o, where the Rays g o, l o, cross the Axis or Perpendicular Ray h o, to be very nigh the Focus of the Eye-Glass ( viz. in our Supposition h o shall be 3,063 Inches) and let the Focal lengths of the Object-Glass and Eye-Glass bear whatever Proportion, the Point o, where g o, l o, shall cross h o, may be determin'd by Prop. V. But then indeed the Rays that fall on the Eye without g o, or within it, do cross the Perpendicular Ray h o, farther from the Eye-Glass, or nigher to the Eye-Glass than g o it self; For they do not proceed from the same Point y; And g o is only the Refraction of the Ray y g. And therefore, unless the dif­ference between the Focal lengths of the Object-Glass and Eye- Glass be Considerable, the Eye may move considerably nigher to the Eye-Glass, or farther from the Eye-Glass than the Point o, and not perceive any Alteration in the Appearance of the Object. But if the said Difference be Considerable, (as it always is in Telescopes) the Eye can move but very little either far­ther from or nigher to the Eye-Glass than o, but it shall perceive a great Alteration in the Appearance (that is, in the Visible Area) of the Object. As shall be manifested more plainly hereafter.

From all which it appears, That the Pupil of the Eye at o, being in the place of the greatest Confusion, where the Rays [Page 159] from all Points in the Object are mixt together, and the Rays from each single Point fall parallel amongst themselves; The Appearance of the Object must needs be Distinct. By Prop. XXVIII. Sec. 3.

Fifthly, I say likewise the Object is magnified. If the naked Eye were in the place of the Object-Glass at y (Tab. 35. f. 2.) T. 35. F. 2. the Object would appear to it under the Angle a y c = f y d. That is, the Object would appear to the Eye at y, under the same Angle as the Distinct Base f e d, were it an Object, would appear to the same Eye at y. Wherefore let us now consider the Distinct Base f e d as the Object. This appears to the Eye at o through the Eye-Glass under the same Angle g o l, as were the naked Eye viewing it at h, by Prop. XXXIII. Make [...] q equal to e h, and draw h f, h d, and q f, q d. The Angle f h d by Prop. XXXIV. is equal to g o l, the Optick-Angle through the Telescope, and the same Angle f h d is equal to f q d. But f q d is much greater than f y d = a y c the Natural Optick-Angle to the Eye supposed at y. and yet much greater than the Natural Optick-Angle would be to the Eye at o. Be­cause o is yet further from the outward real Object than y by the whole length of the Telescope. Wherefore the Object is Magnified.

Lastly, I say the Object appears Inverted. This is manifest from the very Scheme ( Tab. 35. f. 1. T. 35. F. 1.) without farther Ex­plication. For the Object is Inverted in the Distinct Base, and the Eye-Glass does not return it again before it arrive at the Eye, but 'tis painted on the Retina r s t Erect; wherefore by Prop. XXXVIII. Sec. 4, 5. the Object appears Inverted. See also Prop. XXXIX. Sec. 4, 5.

PROP. LI. LEMMA.

If the Eye directly appraoch to or recede from an Object; it shall be, as the Tangent of the Semioptick-Angle of one Station : to the Tangent of the Semioptick-Angle of t'other Station :: So (reciprocally) the Distance of the Eye from the Object in this lat­ter Station: to the Distance of the Eye from the Object in the Former Station.

Tab. 35. f. 3. T. 32. F. 3. Let a b c be an Object, from whose middle Point c erect c e Perpendicular to a b. Let the first Station of the Eye be at e, and its second Station at d; and draw a d, a e; I say therefore, As c e : to c d :: so Tangent of the Angle a d c : to the Tangent of the Angle a e c.

PROP. LII. LEMMA 2.

If the Eye directly approach to, or recede from an Object, its appa­rent Bigness increases or diminishes, at the Tangents of the Semioptick-Angles at one and t'other Station.

This is manifest, for the Eye at d (Tab 35. f. 3. T. 35. F. 3.) sees a c as big as the Eye at e would see z c, by Prop. XXVIII. Sec. 6. Because the Angles a d c, z e c are equal; and Quae sub aequali Apparent Angulo, Aequalia videntur. So that the Eye advan­cing from e, to d, sees a c as much bigger, as if, continuing at e, the Object a c had increased to c z.

PROP. LIII.

The apparent Diametral Magnitude of an Object viewed through the Telescope of Prop. L. Is to the apparent Diametral Magni­tude of the Object viewed by the naked Eye at the Station of the Object-Glass :: As the Focal length of the Object-Glass: to the Focal length of the Eye-Glass.

This is the great Proposition asserted by most Dioptrick Wri­ters, but hitherto proved by none (for as much as I know) [Page 162] they offer indeed Experiments and Methods of Tryal to con­firm the Truth thereof, but proceed no farther.

Vid. Cherubin Diop. Oculaire Part. II. Prop. XXI. LIX. LX. LXII. Kepleri Dioptr. Prop. CXXIV. Galilei Nuncius Sidereus pag. 12. Edit. Lond. 1653. 8vo.

Honoratus Faber in his Synopsis Optica Prop. XLIV. for the Telescope consisting of a Convex Object Glass and Concave Eye-Glass; and in Prop. XLV. for the Telescope consisting of a Convex Object-Glass and Convex Eye-Glass, indeavours at something, which he calls a Demonstration of this Property. But whether that which he there offers will amount to clear Satisfaction, I leave to their Judgements, who shall Read him.

Dechales in Prop. LIV. Lib. II. Dioptr. thinks this Proposi­tion so far from Demonstrable, that he takes it to be False; and says, He never met with any Demonstration thereof, that did not include manifest Paralogisms. Perhaps he may be right in this latter part of his Assertion; but the Reason he gives for his concluding it a False Proposition is manifestly Weak and Erroneous: And that on the account of an inartificial kind of Notion and Method, that he takes for Explicating the Mag­nifying of Telescopes; especially of the Telescope furnish'd with a Concave Eye-Glass, which he explains after his manner in Prop. LIII.

It were needless to inlarge in this matter, I shall therefore pass it over, and hasten to the Demonstration of this Proposition.

Tab. 35. f. 2. T. 35 F. 2. Let the Object-Glass x y z Project the Image of the Object A B C in the Distinct Base f e d, g h l is the Eye-Glass, h e the Focal length of the Eye-Glass, to which let e q be made equal. Draw h f, h d, and q f, q d. And let the Eye at o be placed in the Exteriour Focus of the Eye-Glass. The Rays y f g, y d l, are refracted by the Eye-Glass, and cross at o. Wherefore g o l is the Angle under which the Object ap­pears through the Telescope. But the Angle g o l is equal to [Page 163] the Angle f h d (as well by what foregoes in Prop. L. Sec. 3. & 5. as by Prop. XXXIV.) And the Angle f h d is equal to the Angle f q d.

Wherefore were the naked Eye at y the Station of the Ob­ject-Glass, it would perceive the Object under the Angle f y d = a y c. But now being armed with the Telescope, it sees the Object under the Angle f q d. Let us take their halfs f y e, f q e; and consider the Angle fe, half the Distinct Base, as the Object viewed by the naked Eye at the Stations y and q. I say (by Lemma 2.) The apparent bigness of the Object f e at the Station q : Is to its apparent bigness at the Station y :: As the Tangent of the Angle f q e : To the Tangent of the Angle f y e. But (by Lemma 1.) the Tangent of the Angle f q e : Is to the Tangent of the Angle f y e :: As e y : To e q. Therefore the apparent bigness of the Object f e at the Station q : Is to its apparent bigness at the Station y :: As e y : To e q : that is, As the Focal length of the Object-Glass: To the Focal length of the Eye-Glass. But the Angle g o h, under which the Eye sees half the Object through the Telescope, is equal to the Angle f h e or f q e. Therefore the apparent Dia­metral bigness of an Object viewed through a Telescope: Is to the apparent Diametral Magnitude of the Object viewed by the naked Eye at the Station of the Object-Glass :: As the Focal length of the Object-Glass: To the Focal length of the Eye-Glass. Which was to be Demonstrated.

The same may be declared otherwise. Thus; Tab. 35. f. 2. T. 35. F. 2. Let us suppose the naked Eye at h to view the Object In­verted by means of the Distinct Base f e d; The Inverted Ob­ject shall appear under the Angle f h d (by Prop. XL.) But the Eye at o through the Glass perceives the Inverted Image of the Object under the Angle g o l equal to f h d, (by Prop. XXXIII.) and f h d is equal to f q d, and consequently (as in the foregoing Demonstration) the Proposition is manifest.

[Page 164]I shall now mention the common Method for trying the Truth of this Proposition by Experiment. Having the Focal length of an Object-Glass (for Instance) 144 Inches, and the Focal length of an Eye-Glass three Inches. A Telescope composed of these, shall make the apparent Diamet [...]al Magni­tude of an Object: To the apparent Magnitude of the same Object viewed by the naked Eye :: As 144 : To 3 :: or 48 : To 1. Wherefore, such a Glass is said to Magnifie 48 times in the Diameter of the Object, and 2304 (= square of 48) in the Surface of the Object. The Superficies of like Fi­gures being to each other, as the Squares of their Diameters, or Homologous Sides.

Wherefore from a convenient Scale take one part, and there­with describe a Circle; And from the same Scale take 48 parts, and describe another Circle. Let these two Circles be cut out in Paper, or other Conspicuous Material, and placed at three or four Foot from each other, on a Wall at such a Distance as will require the length between the Glasses in the Telescope but just 147 (= 144 + 3) Inches, to shew these Objects distinctly; Then with one Eye through the Telescope observe the smaller Circle, and at the same time with t'other Eye naked look upon the greater Circle; these two Circles shall appear equal to both Eyes.

Perhaps it may be objected, That the Comparison is not fair between both Appearances. For the Proposition supposes the naked Eye at the Station of the Object-Glass; But this Experiment sets the naked Eye Distant from the Object-Glass the whole length of the Telescope. This would be a mate­rial Objection against this Method of Tryal, were not the Distance of the two Circles from the Eyes vastly greater than the length of the Telescope; so that the Telescopes length may not bear any sensible Proportion thereto. And such we sup­pose it in this Experiment, by advertising that this Distance is [Page 165] to be so great, that the Distance between the Glasses may be no longer than for viewing a Distant Object, viz. the just Aggregate of the Focal lengths of the Glasses, that is, 144 + 3 = 147 Inches, that is, ye + eh = yh.

I shall now give an Example of a Calculation according to this Proposition. Wherefore in Tab. 35. f. 4. T. 35. F. 4. let us take the Moon ABC for our Distant Object; and let us suppose its Diameter to subtend an Arch of a great Circle of Heaven of 30' Minutes. Let the Ray a y d proceed from its upper Limb, b y e from its Centre, c y f from its lower Limb. These cross in the Vertex y, or middle Point of the Object-Glass x y z; making the Angle f y d = a y e, equal to 30 Minutes. Let the Focal length of the Object-Glass e y be given twelve Feet = 144 Inches, or 144,00 Parts: And the Focus of the Eye-Glass h e be given three Inches, or 3,00 such Parts. Let the Distinct Base, wherein the Image of the Moon is Projected by the Ob­ject-Glass be f e d, and draw f h, d h. It is shewn before that the Angle g o h is equal to the Angle f h e.

Wherefore in the Right-angled Triangle f e y, we have e y = 144,00, and the Angle f y e = 15', to find f e = 0,63.

Then in the Right-angled Triangle f e h, we have h e = 3,00, and f e = 0, 63, to find the Angle f h e = 11° 51' 40".

Wherefore the Semidiameter of the Moon, which by the naked Eye would be seen under the Angle of 15' Minutes, is seen through this Telescope under an Angle of 11° 51' 40".

Let us now enquire, whether the Object appearing under an Angle of 15° Minutes, and being afterwards made to ap­pear under an Angle of 11° 51' 40", doth not thereby appear 48 times bigger than naturally. (for so much, by what fore­goes, does this Glass Magnifie).

And for shewing this, let us imagine f e increased 48 times its length 0, 63; And then inquire what Angle f y e would be. Where­fore f e is now supposed = 30, 24 = 48 times 0. 63; Then [Page 166] in the Right-angled Triangle f e y, we have f e = 30, 24, and e y = 144,00, to find the Angle f y e = 11° 51' 40". Which shews that the Semidiameter of the Moon, being made by the Glass to appear under an Angle of 11° 51' 40", is seen by the Eye as big, as if the Semidiameter of the Moon it self were really increased 48 times, and viewed by the naked Eye. Which is the proposed Design of this Calculation.

PROP. LIV. PROBL.

To Determine the Angle received by a Telescope of the foregoing Combination. The Rule is, as the Distance between the Ob­ject-Glass and Eye-Glass: To half the Breadth of the Eye-Glass :: So Radius : To the Tangent of half the Angle received.

Tab. 25. f. 2. T. 35. F. 2. The Distance of the Glasses is h y. Let half the Breadth of the Eye-Glass be g h. Then, as h y : To g h :: So Radius: To the Tangent of the Angle g y h, which is half the Angle g y l, the Angle received. That is, the Eye at o shall perceive no more of the Object, than subtends this Angle before the Object-Glass.

PROP. LV.

Concerning the Apertures of Object Glasses.

By the Aperture of a Glass I mean, that part of the Glass which is left open and uncovered. And this ought to be va­rious [Page 170] according as we would have more or less Light admitted. It also varies according to the various Focal lengths of the Ob­ject-Glasses. For a ten Foot Object-Glass shall bear a greater Aperture than an Object-Glass of one Foot; and a twenty Foot Glass yet greater than a ten Foot Glass.

But at what Rate or Proportion the Apertures of Glasses alter in respect of their lengths, is not yet well setled.

Monsieur Auzout, (Phil. Transact. N. 4. P. 55.) Tells us, that he finds, That the Apertures, which Glasses can bear with Distinctness, are in (about) a Subduplicate Ratio to their lengths: Or as the Square Roots of their lengths. Whereof he intends to give the Reason and Demonstration in his Dioptrica (which we yet want.) But this Ingenious Person should have told us, when he speaks of the Apertures of Glasses, whether he de­signs them for Objects on the Earth or in the Heavens. And if in this latter, whether for the Moon, Mars, Iupiter, or Venus. For each of these Objects will require a different Aperture of the same Glass. Because the Strength of their Light is diffe­rent. For to view Venus there is requisite a much smaller Aperture, than to view the Moon, Saturn or Iupiter.

However till some better Rule can be found for settling the Apertures of Object-Glasses (which at present I shall not pre­tend to) I shall here Present you with Mr. Auzout's Table, as 'tis to be found in the fore-cited Philosophical Transaction, Numb. 4. Noting only, that his Feet are Parisian Feet (which is to the London Foot as 1068: to 1000) and each Inch (which is the ¹¹² part of his Foot) is subdivided into twelve Lines. For it had not been worth our Pains to have reduced the whole Table to our English Measure. Vid. Tab. 36.

I have said before ( Schol. 2. Prop. LIV.) That the Angle received, or Visible Area of an Object, is not Increased or Di­minished by the greater or lesser Aperture of the Object-Glass; all that is effected thereby is the Admittance of more or less [Page]