I. Note.

AS an Abstract from the undoubted Axioms of Geometry, it is generally observed, that in a Rank of numbers, having equal diffe­rence, the second differences of the squares of those numbers are equal; the third differences of the cubes of those numbers are equal: and so in order in the higher powers. Thus, [Page 2]

Observe in the first of these Examples, in the first collum are the numbers of a progressition, having equal difference, to wit, a unite. In the second co­lum, the squares of those numbers. In the third co­lum, the first differences. In the fourth colum, the second differences, to wit, 2, 2, 2. In the second Example, in the first colum are a Rank of numbers, having equal difference, to wit, 2. In the second colum, their squares. In the third colum, the first difference. In the fourth colum, the second diffe­rence.

II. Note.

Hence it follows, that by the help of such diffe­rences the table of squares may be calculated: thus, in the first Example, the sum of 1 and 3, is 4; the square of 2. The sum of 2, 3 and 4, is 9; the square of 3. The sum of 2, 5 and 9, is 16; the square of 4. The sum of 2, 7 and 16, is 25; the square of 5. The sum of 2, 11 and 36, is 49; the square of 7. &c.

III. Note.

Like plain numbers are in the same proportion one to another, that a square number is in, to a square number: Euclide the 26 Proposition of the Eighth Book. Therefore the second difference in such a Rank of plane numbers are equal. Further, what planes and solids are either equal or propor­tionable to such Ranks may be gradually calculated; as in the last.

IV. Note.

In the first Example, in the first colum are the numbers in a Rank having equal difference, to wit, a unite. In the second colum, the cubes of those numbers. In the third colum, the first differences of those cubes. In the fourth colum, the second differences. In the fifth colum, the third differen­ces, to wit, 6, 6. The like in the second Example.

V. Note.

Hence it follows, that the table of cubes may be [Page 4] made thus: In the first Example, 1 and 7, is 8; the cube of 2. The sum of 8 and 19, is 27; the cube of 3. The sum of 18, 19 and 27, is 64; the cube of 4. The sum of 6, 18, 37 and 64, is 125; the cube of 5. The sum of 6, 24, 61 and 125, is 216; the cube of 6. The sum of 6, 30, 91 and 216, is 343; the cube of 7. The like in the second Example.

VI. Note.

Like solid numbers are in the same proportion one to another, that a cube number is in, to a cube number▪ Euclide the XXVII Prop. of the Eighth Book. Therefore the third differences in such a Rank of solid numbers are equal: further, such planes and solids as are either equal or proportio­nable to such Ranks, may be gradually calculated, as in the last.

VII. Note.

If a Rank of Squares, whose Roots have equal differences, be multiplied by any number, the se­cond differences of such a Rank of proucts are e­qual. Let the number multiplying be 10.

In the first colum are the num­bers bearing equal difference. In the second colum are the squares of those numbers. In the third colum the products. In the fourth colum the first differences. In the fifth colum the second differences and they equal.

VIII. Note.

If unto such a Rank of Products, as in the last, there be added a Rank of Cubes, whose Roots are equal to the Roots of the Squares, the third diffe­rences of such a Rank will be equal.

In the first colum are the numbers having equal difference. In the second colum the Products of their squares by a given number. In the third co­lum the cube of the numbers in the first colum. In the fourth colum the sum of the products and cubes. In the fifth colum their first difference. In the sixt colum the second differences. In the seventh co­lum the third differences which are equal.

IX. Note.

Let a constant number be added to a Rank of Products, so that one of the numbers multiplying [Page 6] be a constant number, and the other of the num­bers be the squares of numbers having equal diffe­rence, and this Rank of sums be added to a Rank of cubes, whose roots are the same with the roots of the squares; such a compounded Rank will have their third difference equal. Thus,

In the first colum are the numbers having equal difference. In the second colum is constant number to be added. In the third colum are the Rank of products, that is, the squares of the num­bers in the first colum multiplied by a given num­ber. In the fourth colum are the cubes of the num­bers in the first colum. In the fifth colum are the sum of the numbers in the second, third and fourth colums. In the sixth colum are the first differences of their sums. In the seventh colum are the second differences. In the eighth colum the third differen­ces, and they equal.

X. Note.

In a Rank of numbers, having equal difference, and equal in number; if the third part of the cubes of each of these numbers, be substracted from the products of the squares of each of these numbers, in half the greatest number of that Rank, the re­mainders [Page 7] will be a Rank of numbers, equal to all the squares in the several portions of one fourth of a sphere, whose diameter is equal to the greatest number in that Rank, and the third differences of this Rank of portions are equal; but the first and second differences will increase and decrease, diffe­rently one from another. Thus,

In the first colum are the numbers having equal difference. In the second colum are the pyramides adscribed within the cubes of the numbers in the first colum. In the third colum are the products of the squares of the numbers in the first colum, by half the greatest number in the first colum. In the fourth colum are the differences of the numbers in the second and third colums, that is, all the squares in several portions of one fourth of a sphere, whose diameter is 24. In the fifth colum are the first dif­ferences. In the sixt colum are their second diffe­rences. In the seventh colum are the third differen­ces, and they equal.

XI. Note. The Application or Vse of the Preceeding Notes.

The application or Use may be, to calculate Py­ramides and cones, either the whole or their parts▪ as also to calculate the parabolick and hyperbolick conoides, either the whole, or their frustums; yet also, to calculate the sphere or spheroide, either the whole or their portions or Zones, and that gradually, that is, to find the solidity upon every inch or foot.

XII. Note.To find the solidity of an hyperbolick conoid gra­dually, to wit, upon every three inches.

For the performance of which, take notice of the XVI Prop. of Stereom. Prop. in that Diagram, Let AM equal to AB be 9. Let ML equal to AF, be 6. Let AE be 15: therefore FE will be 9. Let the rest of the construction be as in that Pro­position. Let from M to the first K be 3, whose square is 9, whose half is 4½, the area KHM; which being multiplied by ML, 6; the product will be 27, the prism KHNOLM. Because FE, FC and FL are equal, that is, each of them 9: Therefore, the first pyramid ONXILM will be 9. Then this prism and pyramid being added, will make 36, the whole prism KHXILM, equal to all the squares in the portion KZM. Let from M to the second K be 6, whose square is 36, its half 18, the area KHM; which being multiplied by ML, 6; the product will be 108, equal to the prism KHNOLM. The cube of 6, is 216; a third part is 72, the pyramid ONXIL, this prism and pyramid be­ing added together, the sum will be 180; the prism KHXILM: equal to all the squares in the por­tion KZM. Let from M to A, be 9; its square 81, the half 40½, equal to the area ABM, which being multiplied by ML, 6; the product will be 243: the cube of 9, is 729; a third part thereof is 243; equal to the pyramid FCDEL: this prism and pyramid being added together, is 486; the [Page 13] whole prism ABDELM, equal to all the squares of the portion AZM.

These three portions being obtained, they may be continued by the VIII Note, thus:

For if the third dif­ferences which are e­qual, and in this Ex­ample is 54, be added to the first of the se­cond differences, be­ing 108, it makes 162, and by such additions, the second differences in the fourth colum are made. Further, by adding these second differences to the first of the first differences which is 36, it makes 144, &c. So the numbers in the third colum are made. Yet further, by adding these first differen­ces to the first number in the second colum, the Rank of portions of such a conoid is made.

Then,

By making use of the directions in the first and second Scholiums, the number of Gallons are ob­tained. The parabolick and hyperbolick conoides may well be made use of for Brewers Coppers; the parabolick, when the crown is somewhat blunt; but the hyperbolick conoid when the crown is more sharp.

XIII. Note. To calculate a Sphere gradually, to wit, upon every three Inches.

Consider the XV Prop. of Stereom. Prop. Let ED equal to EF, be 24. The rest of the constru­ction as in that Prop. Let from E, to the first R be 3, whose square is 9; whose half is 4½, the area RXE, which being multiplied by 24, the product will be 108; the prism KHXREF. The cube of 3, is 27; a third part thereof is 9, the pyramid KHOIF; this pyramid taken from the former prism, leaves the prism RXOIFE, 99: equal to all the squares in the portion RQE. From E, to the second R, 6; its square 36, the half 18, which multiplied by 24, makes 432; the prism RXHKFE. The cube of 6, is 216, a third part of it is 72; the pyramid KHOIF: this pyramid being taken from that prism, there rest 360; the prism RXOIFE, equal to all the squares in the portion RQE. Let from E to the third R be 9; its square 81, the half thereof 40½, the area RXE, this area being multiplied by 24, the product will be 972, the prism KHYREF: the cube of 9, is 729, a third part of it is 243, the pyramid KHOIF: this py­ramid being substracted from that prism, the re­mainder is 729; the prism RXOIFE, equal to all the squares in the third portion RQE. Having obtained these three portions, the rest may be found by their third difference, according to the X. Note.

[Page 15]

The numbers in the seventh colum are the third differences, and they equal; the numbers in the sixt colum are the second differences, and are com­posed by substracting the numbers in the seventh from the first and last numbers in the sixt colum; the numbers in the fifth colum are the first diffe­rences, and are composed by adding those num­bers in the sixt colum to the first and last of those in the fifth colum; the numbers in the fourth colum are all the squares in several portions of one fourth of a sphere whose diameter is 24, those por­tions are made by adding the numbers in the fifth colum to the numbers in the fourth, thus, 261, and 99, is 360. 369, and 360, is 729. 423, and 729, is 1152, &c.

Then making use of the first and second scho­lium the number of gallons are obta [...]ned. Or if it be made, as 14, is to 11, so is 54, to a fourth [Page 16] number, with that fourth number proceede to make tables of the second and first differences, and then the table of portions it selfe. Every sphere hath its third differences equall. To find the third difference, doe thus. Cube one of the equal segments of the axis and multiply that cube by 2, and that product will be the third difference, thus, the cube of three is 27, which multiplyed by 2, the product is 54; the third difference of all the squares in one fourth of a sphere. Here note, that it is to be understood, that the axis of the sphere is equally divided into an equal number of segments; so then, if the number of segments in the semiaxis, less by one; be multiplyed by the third difference, it gives the first of the second differences. Thus, the number of segments in the semiaxis is 4, then 4 less 1, is 3; which being multiplyed by 54, the product is 162: the first of the second differences.

To find the third difference in one fourth of all the squares in a spheroid, do thus: The axis being divided as above in the sphere; cube the difference betwixt two Segments, which being multiplyed by 2, makes a product; then, as the square of the semiaxis, is to the square of the other semidiameter; so is that former product to a fourth number, which will be the third diffe­rence. For the second differences, use the Rules given for the sphere.

XIV. Note.

To calculate a pyramid or cone gradually. To find the third difference in a pyramid work thus, [Page 17] the Altitude of the pyramid being equally divided▪ cube the difference of the two segments, which being doubled, makes a number; then, as the square of the Altitude of the pyramid, is to the area of the base of that pyramid; so is that for­mer number, to the third difference of that pyramid.

To find the second differences in a pyramid: As the difference of two of the segments of the Altitude, is to the following segment; so is the third difference, to the second difference answer­able to that segment.

To find the first differences in a pyramid. Cube two of the segments, and take a third part of their difference. Then, as the square of the Altitude of the pyramid; is to the area of the base of that pyramid; so is that former difference; to the first difference answerable to those two segments.

Let there be a pyramid whose Altitude is 10, and one side of the base is 40, and the other side 5; therefore the area of the base is 200. Let the Altitude be divided into five equall parts, and to calculate accordingly. To find the third differ­ence, the cube of 2, is 8; whose double is 16. Then, as 100 the square of the Altitude, is to 200 the area of the base; so is 16, to the third difference 32. To find any of the second differ­ences at demand, to find the second difference an­swerable to 8. As 2, the difference betwixt the segments 6, and 8, is to 8; so is the first difference 32, to 128 the second difference answerable to 8. The second differences are in proportion one to another, as their answering segments; as 2, is to [Page 18] 3▪2; so is 8, to 128. To find any of the first dif­ferences, cube the two Segments, to wit, 2 and 4, and the cubes will be 8 and 64; then take 8 from 64 and the Remainder is 56, a third part is 18⅔. then, as the square of the Altitude 100, is to the area of the base 200; so is 18⅔, to 37⅓, the first difference, answering to 2 and 4. Then by a continuall adding of the third difference to the second differences they are made, and by ad­ding the first of the second differences to the first of the first differences and so in order the first differences are made. Lastly by adding the first differences the Segments of the pyramid, are made according to the III. Note; or thus.

The numbers in the fifth colum are the third differences, the first number in the fourth colum being found by the Rule before given, all the numbers in that fourth colum may be made by adding the third difference, thus, to 32 adde 32, the summe is 64. adde 32, to 64; the summe is 96. adde 32, to 96; the summe is 128. The first number in the third colum being found by the Rule above, then 5⅓ added to 32; the summe is 37⅓. [Page 19] adde 64, to 37⅓; the summe is 101⅓. adde 96, to 101⅓, the sum is 197⅓. adde 128, to 197⅓; the sum is 325⅓. further, adde the first of the third colum, to the first of the second colum; thus, adde 5⅓, to 0; the sum will be 5⅓, adde 57⅓, to 5⅓, the sum is 42⅔, adde 101⅓, to 42⅔; the sum is 144. adde 197⅓, to 144; the sum is 341⅓, adde 325⅓, to 341⅓; the sum is 666⅔.

If it be to calculate a cone whose diameters of the base are 40 and 5. Let it it be made, as 14, is to 11; so is 32, to the third difference of the same cone. Then proceede with the third dif­ference to make the second and first; and lastly, the table it self.

XV. Note.

The calculation of frustum pyramides whose bases are unlike, To the performance of which consider the third case of the second proposition of Stereom. Prop.

Every such solide hath its third differences equall, but the second and first differences will be complicated according to the IX. Note.

To find the third difference proper to the pyramid BCDHF, Let the construction and numbers be the same as in that diagram, and let it be to calculate it upon every two inches, thus. The cube of 2, is 8; the double thereof is 16, Then, as the square of the Altitude 40, that is 1600, is to the area of the base BCDH, 336; so is 16, to 336/100. by the Rule delivered in the [Page 20] 14 Note, the first of the second differences is 336/100. and the first of the first differences is 56/100.

The solide HDEGVF hath its second differ­ences equall by the VII. Note.

To find its first and second differences. The square of 2, is 4. which multiplyed by FV, 26; the product will be 104. then, as 40 the Altitude, is to HD, 28; so is 104, to 7280/100. the second difference. Therefore the first of the first differ­ences will be 3640/100.

To find the second differences of the solide ABHOIF the square of 2 is 4, which multi­plyed by IF, 30; the product is 120, then, as 40, the Altitude; is to OA, 12: so is 120, to 36, the second difference. Therefore the first of the first differences are 18. For the complication of these differences.

Rejecting the denominators they may be writ­ten Thus,

Because the denominators are Rejected, there­fore the two last figures toward the Right hand are decimals.

[Page 21]

[Page 22]The construction of the table may be thus; the numbers in the first colum are the third dif­ferences▪ The first number in the fourth colum is the complicated second difference, and the other number in that fourth colum are made thus, to the first 11216, adde 336; the sum is 11552. Then to that 11552, adde 336; the sum is 11888, &c.

The first number in the third colum is compli­cated from the first complicated difference and a parallelipepidon whose base is the plane RIFV, and the Altitude the first Segment of the Altitude of the frustum, thus, the plane RIFV, is 780; which being doubled is 1560; then, 156000 more 5496 is 161496; the first of the first differences, then 161496 more 11216, is 172712. Further, 172712 more 11552, is 184264. Yet further 184264 more 11888, is 196152, &c.

The numbers in the second colum are made thus, the first number in the second colum, is the same as the first number in the third colum, then, 161496 more 172712, is 334208, and 334208 more 184264, is 518472, &c.

Then makeing use of the first and second scho­lium, the quantity of Liquor that such vessels contain may easily be obtained.

XVI. Note.

To calculate Elliptick solides whose bases are unlike. The calculation of such solides are the same as in the 15, note for if the first, second and third complicated differences be found, then make­ing use of this propotion as 14, is to 11; so is 336; to the third difference.

[Page 23]And

As 14, is to 11; so is 11216, to the first of the second differences.

Further

As 14, is to 11; so is 161496, to the first of the first differences, then proceede to make the table it self, as in the 15 note. Or make use of the secund scholium of the 11 note and you will have the quantity in Gallons.

Or

Such Elliptick solides may be calculated by the 12 note: for every such Elliptick solide is equall to a frustum hyperbolick conoide whose circular bases of the conoide, are equall to the Elliptick bases of the Elliptick solide; and the Altitude of one frustum is equall to the Altitude of the other.

XVII. Note.

Every hyperbolick conoid hath its third diffe­rences equal.

To find the third, second and first differences in an hyperbolick conoid, and consequently to calculate that conoid gradually. In the foremen­tioned diagram of the 17. prop. Stereom. Prop. Let GM, the Transverse diameter be 12. ML, the parameter 6. MA, the axis of the conoid 24.

XVIII. Note.

XIX. Note. In a Rank of numbers having equal differences.

Let the first term in the Rank be Z, its square ZZ. the second term 2Z, its square 4ZZ, there­fore the first of the first differences is 3ZZ, the third term in that Rank 3Z, its square 9ZZ, then 9ZZ, Less 4ZZ, the second of the first differences 5ZZ, therefore 5ZZ Less 3ZZ the second difference will be 2ZZ.

Further,

The fourth term in that Rank is 4Z, its square is 16ZZ, then 16ZZ Less 9ZZ the third of the first differences 7ZZ; again, 7ZZ Left 5ZZ the second difference is 2ZZ.

Hence it follows,

That the second difference is equall to the square of the first term doubled.

Or also,

The second difference is equall to the squar [...] [Page 32] of the difference of two of the terms, (in order taken) doubled.

By the same method we find that the third difference in a Rank of cubes are equall, and the third difference is equal to the first term multi­plyed by 6.

Or,

The third difference is equal to the cube of the difference of two of the terms, taken in order, multiplyed by 6.

The index and equal difference, of every power agrees; to wit, the index of the square is 2, and the second differences are equal. The index of the cube is 3, and the third differences are equal. The index of the square squared is 4, and the fourth differences are equal. &c.

The equal difference of every power, is com­plicated from the index of that power, and the equal difference of the next Lesser power.

Let the Rank be in naturall order, Thus; 1, 2, 3, 4. &c.

The indices of the powers, Thus.

A unity the equal difference in that naturall Rank, whose square is 1, which multiplyed by 2 the index of the square the product is 2, the equal difference in the squares. 3, the index of the cube multiplyed by 2 the equal difference in the squares, [Page 33] the product is 6, the equal difference in the cubes▪ 4 the index of the square squared multiplyed by 6 the product is 24 the equall difference in the square squared, &c.

If the Rank be in order thus, 2, 4, 6, 8, &c.

2 the equal difference of this Rank whose square is 4; multiplyed by 2 the index of the square the product is 8; the equal difference of the squares in such a Rank. Because the equal diffe­rence of the Rank is 2, therefore the indices are to be doubled, &c.

And the equall difference of the powers in such a Rank will be 8, 48, 384, &c.

XX. Note. For the more easier calculation of the second sections of the sphere and spheroid; worke, Thus.

From the double of the superficies of the triangle BZN, substract the superficies of the triangles BZGD and NZPA, the Remainder will be the superficies of the triangles BZPA and NZGD, the areas of these two triangles being substracted from the area of the traingle NZB, the Remainder will be the superfice of the triangle ZGDAP.