GAUGING PROMOTED. AN APPENDIX TO Stereometrical Propositions. By ROBERT ANDERSON. LONDON, Printed by I. W. for joshua Coniers, at the Raven in Ducklane, 1669. Gauging Promoted AN APPENDIX TO Stereometrical Propositions. 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, In Squares 1 1 1 3 2 4 2 5 3 9 2 7 4 16 2 9 5 25 1 2 3 4 2 1 1 8 3 9 8 16 5 25 8 24 7 47 8 32 9 81 1 2 3 4 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. etc. III. Note. Like plain numbers are in the same proportion one to another, that a square number is in, to a square number: Euclid 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. 1 1 7 2 8 12 19 6 3 27 18 37 6 4 64 24 71 5 125 1 2 3 4 5 1 1 26 3 27 72 98 48 5 125 120 218 48 7 343 186 386 9 729 1 2 3 4 5 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 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▪ Euclid 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. 0 0 00 10 1 1 10 20 30 2 4 40 20 50 3 9 90 10 70 4 16 160 20 90 5 25 250 1 2 3 4 5 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. 0 00 0 00 11 1 10 1 11 26 37 6 2 40 8 48 32 69 6 3 90 27 117 38 107 6 4 160 64 224 44 151 5 250 125 375 1 2 3 4 5 6 7 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 sixth 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 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, 1 8 10 1 19 37 2 8 40 8 56 32 69 6 3 8 90 27 125 38 107 6 4 8 160 64 232 44 151 5 8 250 125 383 1 2 3 4 5 6 7 8 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 columns. 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 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, 0 00 00 00 99 3 09 108 99 162 261 54 6 72 432 360 108 369 54 9 243 972 729 54 423 54 12 756 1728 1152 00 423 54 15 1125 2700 1575. 54 369 54 18 1944 3888 1944 108 261 54 21 3087 5292 2205 162 96 24 4608 6912 2304 1 2 3 4 5 6 7 In the first colum are the numbers having equal difference. In the second colum are the pyramids ascribed 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 columns, 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 sixth colum are their second differences. In the seventh colum are the third differences, and they equal. XI. Note. The Application or Use of the Preceding Notes. The application or Use may be, to calculate Pyramids and cones, either the whole or their parts▪ as also to calculate the parabolic and hyperbolic 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. To find the solidity of a parabolic Conoide upon every two inches. To do which, consider the Diagram of the 18 Prop. of my Stereometrical Prop. Let PA be 16; ARE 12; therefore AV or PH will be 9; for it ought to be as PA, is to ARE; so is ARE, to AU. Let the axis AP be divided into eight equal parts; viz. 2, 4, 6, 8, 10, 12, 14, 16. Let there be planes drawn parallel to the base, through every one of these divisions, though in the Diagram there is not so many. From P to the first Q suppose to be 2, its square 4; the half thereof 2, which multiplied by 9 equal to PH, the Product will be 18; that is, the Prism QZGIHP; equal to all the squares in the portion of the conoid QOP. Let from P, to the second Q be 4, its square 16, the half is 8; which multiplied by 9, the Product is 72; equal to the Prism QZFIHP, equal to all the squares in the portion of the conoid QOP. Let from P to the third Q be 6, its square 36, the half of it is 18, which multiplied by 9, the product is 162; the Prism QZEIHP; equal to all the squares in the third portion of the conoid QOP. Having obtained two portions, the rest may be obtained thus: having obtained the second difference, which is 36, we may proceed to find the rest by the Seventh Note; thus, add 36 to 54, and it makes 90: which added to 72, the sum is 162, equal to all the squares in that portion, and so in order; 36, 90 and 162; the sum is 288. 36, 126 and 288; the sum is 450. 36, 162 and 450, the sum is 648, etc. 0 00 8 2 18 36 54 4 72 36 90 6 162 36 126 8 28 36 62 10 45● 36 198 12 648 36 234 14 882 36 270 16 1152 1 2 3 4 In the first colum are the parts of the altitude of the conoid. In the second colum are all the squares in several portions of one fourth of a conoid. In the third colum the first differences. In the fourth colum the second differences. These portions in the second colum may be reduced to circular portions, thus, as 14 is to 11; so are all the squares in these portions to the portions themselves. I. Scholium: The use of this gradual calculation may be thus: Suppose a Brewer's Copper be in form of a parabolic conoid; the quantity of liquor therein contained may be found, thus, having calculated a table upon every inch, or two inches, or as is thought convenient; then having a strait Ruler divided equally into inches, putting the Ruler into the liquor to the bottom of the Copper, see how many inches of the Ruler is wet; with the number of wet inches enter the first colum of your table, and in the next colum are the number of cubick inches which that portion contains; the number of cubick inches thus found, being divided by the number of cubick inches in a Gallon, the quotient shows the number of Gallons in that portion of the Copper. II. Scholium: To compose several works into one. As 14, is to 11; so are all the squares in one fourth of the conoid, to one fourth of the conoid itself. because this one fourth ought to be divided by the number of cubick inches in a Gallon, suppose it 288, to show the number of Gallons in each portion, we may multiply 14 by 288, that is 4032. Then as 4032 is to 11; so are all the squares in one fourth of the conoid, to the Gallons in that one fourth. Further, because this one fourth ought to be multiplied by 4, to reduce it to a whole conoid; therefore, divide the constant divisor, that is, 4032, by 4, and it will be 1008. Then, as 1008, is to 11; so are those several portions in the second colum of the last table, to the number of Gallons in those several portions of a parabolic conoid. By such compositions may the Practitioner compose constant divisors or dividends, which will much breviate the work; this is only for an Example. Every parabolic conoid hath its second differences equal. To find the second differences, work thus, Square one of the equal segments of the axis, and multiply that Square by the Parameter, that product will be the second difference. In this Example, the equal segment of the axis is 2, the square of it 4; which multiplied by the Parameter 9, the product is 36, the second difference. Half of the second difference, is always the first of the first difference. Half 36 the second difference, is 18, the first of the first difference, etc. Here note, this 36 is the second difference of one fourth of all the squares in a parabolick conoid; if 36 be multiplied by 4, it makes the second difference 144; whose half is 72, the first of the first differences. Or, the first differences are found by taking half the difference of the squares of any two segments, which multiplied by the Parameter, thereby the first differences are obtained. Thus, to find the first difference answerable to the Segments 6 and 8; the Square of 8, is 64; the Square of 6, is 36; the difference of those Squares is 28, whose half is 14; which multiplied by the Parameter 9, the product is 126; the first difference answerable betwixt 6 and 8. XII. Note. To find the solidity of an hyperbolic 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 OF, be 6. Let A 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 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: 0 00 36 3 46 108 144 54 6 180 162 306 54 9 486 216 522 54 12 1008 270 792 54 15 1800 324 1116 54 18 2216 378 1494 54 21 4410 432 1926 24 6336 1 2 3 4 5 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, etc. 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 parabolic and hyperbolic conoides may well be made use of for Brewer's Coppers; the parabolic, when the crown is somewhat blunt; but the hyperbolic 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 OF, 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. 0 00 0 0 99 3 09 108 99 162 261 54 6 72 432 360 108 369 54 9 243 972 729 54 423 54 12 576 1728 1152 00 423 54 15 1125 2700 1575. 54 369 54 18 1944 3888 1944 108 261 54 21 3087 5292 2205 162 99 24 4608 6912 2304 1 2 3 4 5 6 7 The numbers in the seventh colum are the third differences, and they equal; the numbers in the sixth colum are the second differences, and are composed by substracting the numbers in the seventh from the first and last numbers in the sixth colum; the numbers in the fifth colum are the first differences, and are composed by adding those numbers in the sixth 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, etc. Then making use of the first and second scholium the number of gallons are obtained. Or if it be made, as 14, is to 11, so is 54, to a fourth number, with that fourth number proceed to make tables of the second and first differences, and then the table of portions itself. Every sphere hath its third differences equal. To find the third difference, do 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 multiplied 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 multiplied by the third difference, it gives the first of the second differences. Thus, the number of segments in the semiaxis is 4, than 4 less 1, is 3; which being multiplied 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 multiplied 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, 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 equal 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 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 continual 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. 0 0 5⅓ 2 5⅓ 32 37⅓ 32 4 42⅔ 64 101⅓ 32 6 144·· 96 197⅓ 32 8 341⅓ 128 325⅓ 10 666⅔ 1 2 3 4 5 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 add 32, the sum is 64. add 32, to 64; the sum is 96. add 32, to 96; the sum is 128. The first number in the third colum being found by the Rule above, then 5⅓ added to 32; the sum is 37⅓. add 64, to 37⅓; the sum is 101⅓. add 96, to 101⅓, the sum is 197⅓. add 128, to 197⅓; the sum is 325⅓. further, add the first of the third colum, to the first of the second colum; thus, add 5⅓, to 0; the sum will be 5⅓, add 57⅓, to 5⅓, the sum is 42⅔, add 101⅓, to 42⅔; the sum is 144. add 197⅓, to 144; the sum is 341⅓, add 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 proceed with the third difference to make the second and first; and lastly, the table itself. XV. Note. The calculation of frustum pyramids whose bases are unlike, To the performance of which consider the third case of the second proposition of Stereom. Prop. Every such solid hath its third differences equal, 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 14 Note, the first of the second differences is 336/100. and the first of the first differences is 56/100. The solid HDEGUF hath its second differences equal by the VII. Note. To find its first and second differences. The square of 2, is 4. which multiplied 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 solid ABHOIF the square of 2 is 4, which multiplied 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. 1 2 3 ●6/100 336/100 3●6/100 in the pyramid BCDHF 3640/100 7280/100 in the prism HGEDFV 18 36 in the prism ABHOIF 5496/100 11216/100 336/100 their sum. Rejecting the denominators they may be written Thus, 5496 11216 336 Because the denominators are Rejected, therefore the two last figures toward the Right hand are decimals. 0 161496 2 161496 11216 172712 336 4 334208 11552 184264 336 6 518472 11888 196152 336 8 714624 12224 208376 336 10 923000 12560 220936 336 12 1143936 12896 233832 336 14 1377768 13232 247064 336 16 1624832 13568 260632 336 18 1885464 13904 274536 336 20 2160000 14240 288776 336 22 2448776 14576 303352 336 24 2752128 14912 318264 336 26 3070392 15248 333512 336 28 3403904 15584 349096 336 30 3753000 15920 365016 336 32 4118016 16256 381272 336 34 4499288 16592 397864 336 36 4897152 16928 414792 336 38 5311944 17264 432056 40 5744000 1 2 3 4 5 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, add 336; the sum is 11552. Then to that 11552, add 336; the sum is 11888, etc. 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, than 161496 more 11216, is 172712. Further, 172712 more 11552, is 184264. Yet further 184264 more 11888, is 196152, etc. 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, etc. Then making use of the first and second scholium, the quantity of Liquor that such vessels contain may easily be obtained. XVI. Note. To calculate Elliptic 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 making use of this propotion as 14, is to 11; so is 336; to the third difference. 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 proceed to make the table itself, as in the 15 note. Or make use of the second scholium of the 11 note and you will have the quantity in Gallons. Or Such Elliptic solides may be calculated by the 12 note: for every such Elliptic solid is equal to a frustum hyperbolic conoide whose circular bases of the conoide, are equal to the Elliptic bases of the Elliptic solid; and the Altitude of one frustum is equal to the Altitude of the other. XVII. Note. Every hyperbolic 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. To calculate the solidity of this conoid upon every three Inches. To find the third difference of this conoid. Take the difference of two of the Segments, to wit, 3; whose cube is 27: whose double is 54. Then, as GM, 12; is to ML, 6: so is 54, to 27. The third difference of all the squares in one fourth of that conoid proper to that pyramid FCDEL. By the Rule in the last note the first of the second differences is 27. For the first of the first differences, work thus; take the first Segment which is 3; whose cube is 27; a third part is 9, then, as GM, 12; is to ML, 6: so is 9, to 4½. the first of the first differences proper to the pyramid FCDEL. The second and first differences of all the squares in the fourth of this conoid, is complicated from the second and first differences of the pyramid FCDEL, and the second and first differences of the prism ABCFLM. Every such prism hath it second difference equal. To find the second and first difference of the prism ABCFLM. Square the difference of two of the Segments of the axis, to wit, 3▪ that is 9, which being multiplied by the parameter ML, 6; the product is 54, the second difference. The first of the first differences of every such prism is half of the second difference; therefore the first of the first differences is 27. To complicate these differences. 1 2 3 differences 4½ 27 27 in the prism FCDEL. 27 54 in the prism ABCFLM. 31½ 81 27 in all the squares of one fourth of that hyperbolic conoid. 0 0 31½ 3 31½ 81 112½ 27 6 144 108 220½ 27 9 364½ 135 355½ 27 12 720 162 517½ 27 15 1237½ 189 706½ 27 18 1944 216 922½ 27 21 2866½ 243 1165½ 24 4032 1 2 3 4 5 In the first colum are the third differences. In the fourth colum the second differences. In the third colum the first differences. In the second colum the portions of all the squares of one fourth of an hyperbolic conoid, upon every three inches, whose Transverse diameter is 12, and parameter is 6, and axis is 24. The construction of this table is the same as the former; thus, 81 more 27; is 108. more 27; is 135. more 27; is 162. etc. 31½. more 81; is 112½. more 108; is 220½. etc. 0 more 31½; is 31½. more 112½; is 144. more 220½. is 364½. more 355½; is 720. Here remember that the Transverse diameter is found, by the 9 of the 23 Proposition. of Stereom. Prop. Also the parameter found by the converse of the first part of the 11 Prop. of Stereom. Prop. The parameter of the parabolike conoid is found, by the converse of the 9 Prop. Of Stereom. Prop. XVIII. Note. Cautions Concerning Reduction. 1 If it be to calculate pyramids whether Regular or Irregular, whole or frustums; the third, second and first differences are to be found as above: then Reduce those differences into Gallons and parts of a Gallon, or Barrels, or parts of a barrels; Thus Suppose 288 cubick inches make one Gallon, and 36 Gallons make one Barrel. Then, If the measure be taken in inches, divide the third, second and first differences by 288, and so there will be three quotients in Gallons or parts of a Gallon, then with those three quotients proceed to make the table of solid Segments, and that table will be in Gallons or parts of a Gallon. If it be to calculate a table in Barrels multiply 288 by 36 and the product will be 10368 the number of cubick inches in one Barrel. Then divide the third, second and first differences by 10368, there will be three quotients in Barrels or parts of a Barrel: Then with these three quotients proceed to make the table of solides Segments. That table being so made will be in Barrels or parts of a Barrel. 2▪ To calculate Cones and Elliptic solids, whether the whole or their frustums. Having found their third second and first differences, as above, and it be to calculate them in cubick inches, Let it be made as 14, is to 11; so is the third difference, to a fourth, And, As 14, is to 11; so is the second difference, to a fourth, Further, As 14, is to 11; so is the first difference, to a fourth with these three number thus found, proceed to make the table of solid Segments, and that table will be in cubick inches. To calculate these solids in Gallons. Multiply 14 by 288 the product will be 4032. Then, As 4032, to 11; so is the third difference, to a fourth. And, As 4032, to 11; so is the second difference, to a fourth. Further, As 4032, to 11; so is the first difference, to a fourth. With these three numbers thus found, proceed to make the table of solid Segments. So that table will be in Gallons. To calculate these solids in Barrels. Suppose 288 cubick inches makes one Gallon, and 36 gallons makes one Barrel, then multiply 288, 36 and 14 one into another and they make 145152. Then, As 145152, is to 11; so is the third difference, to a fourth. And, As 145152, is to 11; so is the second difference, to a fourth. Further, As 145152, is to 11; so is the first difference, to a foruth. With these three numbers thus found make the table of solid Segments: that table will be in Barrels. III. Having found the third, second and first differences of all the squares of one fourth of a sphere, spheroid and hyperbolic Conoid, as in the 12 and 13 notes and the second and first differences of all the squares of one fourth of a parabolic conoid as in the 11 note: they may be Reduced to Circular differences. Thus▪ As 14, is to 11; so is the third difference, to a fourth. And, As 14, is to 11, so is the second difference, to a fourth. Further, As 14, is to 11; so is the first difference, to a fourth. With these numbers thus found make a table, of solid Segments of cubical inches of one fourth of any of these solids. These solid Segments ought to be multiplied by four, to reduce them to solid Segments of a whole sphere, spheroid, hyperbolic and parabolic conoid: but to shun that work divide 14, by four, and then find the new differences; but because 14 cannot be just divided by four, therefore divide 14, by two, and multiply 11, by two, and then work; Thus, Then, As 7, to 22; so is that third difference, to a fourth. And, As 7 to 22; so is that second difference, to a fourth. Further, As 7, to 22; so is that first difference, to a fourth. With these numbers thus found, proceed to make tables as is taught in those Notes: tables so made, will be tables of solid Segments of those solids, in cubick inches. To calculate these solids in Gallons. Multiply 288 by 14, whose product is 4032; one fourth thereof is 1008; Then, As 1008, is to 11; so is the third difference, to a fourth. And As 1008, is to 11; so is the second difference, to a fourth. Further, As 1008, is to 11; so is the first difference, to a fourth. Tables being made, with numbers thus found; according to the former directions in the sphere, spheroid, hyperbolic and parabolic conoids, will be tables of solid Segments of a whole sphere, spheroid, hyperbolic and parabolic conoid, in Gallons or parts thereof. To calculate these solids in Barrels. Multiply 4032 by 36, the product will be 145152, one fourth thereof will be 36288; Then, As 36288, is to 11; so is the third difference, to a fourth. And, As 36288, is to 11; so is the second difference, to a fourth. Further, As 36288, is to 11; so is the first difference, to a fourth. Tables being made, with numbers thus found, according to the former directions, will be tables of solid Segments in Barrels. etc. Then, Using a Rod or Ruler equally divided into inches as in scholium the first, the number of Gallons or Barrels may speedily be obtained. As for the just magnitude of the Gallon, it i● not my business to dispute; that being determined by custom or Authority: I took 288 only for Example sake. 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, Lesle 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 equal to the square of the first term doubled. Or also, The second difference is equal to the square 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 equal, and the third difference is equal to the first term multiplied by 6. Or, The third difference is equal to the cube of the difference of two of the terms, taken in order, multiplied 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. etc. 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 natural order, Thus; 1, 2, 3, 4. etc. The indices of the powers, Thus. 1 2 3 4 5 Z ZZ ZZZ ZZZZ ZZZZZ A unity the equal difference in that natural Rank, whose square is 1, which multiplied by 2 the index of the square the product is 2, the equal difference in the squares. 3, the index of the cube multiplied by 2 the equal difference in the squares, the product is 6, the equal difference in the cubes▪ 4 the index of the square squared multiplied by 6 the product is 24 the equal difference in the square squared, etc. If the Rank be in order thus, 2, 4, 6, 8, etc. 2 the equal difference of this Rank whose square is 4; multiplied 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, etc. And the equal difference of the powers in such a Rank will be 8, 48, 384, etc. XX. Note. For the more easier calculation of the second sections of the sphere and spheroid; work, Thus. From the double of the superficies of the triangle BZN, subtract 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. FINIS. By john Baker, living in Barmonsey-street in Southwark, over against the Princes-Armes, is Taught Arithmetic, both in whole numbers and fractions, Decimal, Logarithmetical and Algebraical, Geometry, Trigonometry, Astronomy, the use of Globes, Navigation, Measuring, Gageing, dialing, etc. Also the Construction and use of all the usual lines put upon Rules or Scales, He also teacheth how to find the (Length and) Spreading of a Hip-rafter, only by a Line of Chords of singular use for Carpenters, a way not as yet vulgarly known amongst Workmen. Faults Escaped in the Impression of Stereometrical Propositions. Page 1, line 18, for and Z Read and H. p. 10, l. 23, for 56, r. 58. p. 34, l. 25, after RI put. p. 43, for 297232, r. 297432. p. 45, for 18, r. V432▪ p. 46, l. 1. for XVI, r. XVII. and l. 21, for A, 16; r. OF, 6; p. 48, l. 6, for 634. r. 624, p. 51. l. 22, for diameter, r. semidiameter. p. 58▪ l. 25, for ZB, r. XB▪ p. 63, l. 1, for XIII. r. XXIII. p. 100 l. 17, for parameter, r. diameter. p. 102 and 103 for as 4 to 3, r. as 3 to 2. p, 105. l. ult. for Z+2▪ r. Z-2. p. 106, l. 4, for Z = ● r. Z-⅗. and l. 25, for 89, r. 98. p. 7 against 12 in the first col. in the sec. r. 576. and in the fifth colum f. 96 r. 99 p. 13. in the second col. f. 46. r. 36. and in the same col. f. 2216, r. 2916.