a 1 equal to 2 X 3 X 4. Therefore the content of the smaller is -th 2 X3 X4 part that of the larger. But the larger contains a number of cubic units equal to 5 X 10 X 17, since A E, AF, AG, contain 5, 10, and 17 lineal 5 X 10 x 17 units, therefore the smaller contains cubic units, or a x 10 2 X 3 X 4 x 17, or 24 X 3}, X 4+ cubic units. Hence the Prop. is true when the dimensions are fractional. Cor. 1. Hence a cube, whose edge contains a units, contains a Xa xa cubic units; therefore a cube, whose edge is 12 inches, or 1 foot, contains 12 X 12 X 12 or 1728 cubic inches; i.e. I cubic foot is equal to 1728 cubic inches. In the same way the other parts of solid measure may be proved. Cor. 2. Hence a rectangular parallelopiped, whose edges are 12 in. 12 in. I in, containing 144 cubic inches, is one-twelfth of a cubic foot: and if a, 6 be numbers of feet in two edges, and c a number of inches in the third, the content of the parallelopiped is a X 12 X 6 X 12 X c cubic inches, or a XbXcX 144 cubic inches, or a X 6 X c twelfths of a cubic foot. But a X b is the number of square feet in a rectangle whose sides contain a, and b lineal feet: therefore a number of square feet in a rectangle multiplied by a number of lineal inches gives the number of cubic primes in the solid, whose base is the rectangle, and height the inches. Also 6 X c is the number of superficial primes in a rectangle whose sides are b feet and c inches; therefore a number of superficial primes in a rectangle multiplied by a number of lineal feet gives the number of cubic primes in a solid whose base is the rectangle, and height the feet. In the ordinary mode of speaking, square feet multiplied by lineal inches give cubic primes; and superficial primes multiplied by lineal feet give cubic primes. Similarly it may be shown that superficial primes multiplied by lineal inches, or superficial seconds by lineal feet, give cubic seconds. Cor. 3. Hence the method of finding the solid content of a rectangular parallelopiped by Cross Multiplication is evident, the dimensions being given in feet, inches, &c. For the content of any parallelopiped (as that in fig. 2,) is evidently the sum of the contents of those whose base is E F, and heights AD, DH, HK, KG. And the contents of these are equal to the sum of the contents of others whose bases are parts of E F, and heights as before. Therefore if these contents be determined, (as they may be by multiplying the square units in the bases by the lineal units in the heights,) and be added together, the sum will be the content of the parallelopiped. a APPENDIX 1.--To explain the Rules of Reduction. The rules for converting numbers of one denomination to another appear axiomatical, when an exact nnmber of one is contained in one of another. Thus, since £1 is equivalent to 20s. it is evident that any sum of money will be equivalent to 20 times as many shillings as pounds; and therefore that, if pounds are to be reduced to shillings, we must multiply by 20, and if shillings are to be converted into pounds, we must divide by 20. And similar reasoning may be used in all other cases of the same kind. But if no exact number of the one denomination be contained in one of the other, the question is evidently one of finding how many times one of the new denomination is contained in the given quantity, or of finding the ratio of the given quantity to the unit of the given denomination. The explanation therefore of the rule is by Prop. 45. 2.- To prove the Rule for conversion of shillings, pence, and farthings, into decimals of a pound. 1s. = 2£. = .050£. = .025£. ļd =of 11d. = .001047£. therefore for every pair of shillings 1 must be placed in the first place of decimals; and for an odd shilling 50 in the 2nd and 3rd places : also for every farthing besides the shillings there must be l in the 3rd place, and, since for 6d. there must be 25 in the 2nd and 3rd places, 1 extra must be added for 6d. Also it appears that for every farthing above the last sixpence, 4 must be put iu the 4th and 5th places, which for 6 farthings or 14d. would become 24, but, since 25 appears in the 4th and 5th places for every 1}d. 1 extra must be added for every 6 farthings. Lastly for every farthing above the last six there will appear in the 6th and following places the decimal figures equivalent to the vulgar fraction ; therefore the figures in these places will be found by converting the fraction, whose denominator is 6, and numerator the number of farthings above the last six, into a decimal. Conversely, the approximate value of a decimal of a pound may evidently be obtained by this Rule :- Take a pair of shillings for every one, in the 1st place, and an odd shilling for 50 (if there be 50) in the 2nd and 3rd places. Also for every one besides in these places take one farthing, subtracting 1 if the number be greater than 24. By converting the decimal into farthings, which is easily done by multiplying by 1000 — 40, this approximation may be corrected. 3.- To explain the Rules of Practice. The cost of a quantity of goods at a given price is equal to the sum of the costs at other prices, which are together equal to the given price; or is equal to the sum of the costs of parts of the goods. Also the cost at any price, which is an exact part of another price, is the same part of the cost at this latter price, and may therefore be obtained from the latter by Divi. sion. And the cost of any quantity of goods, which is an exact part of another quantity, is the same part of the cost of this latter quantity, and may therefore be obtained from the latter by Division. Hence if the price given be separated into parts, each of which is an exact part of some price, the cost at which is known, the costs at each of these may be found in succession by division, and their sum will be the total cost. Or if the quantity of goods given be separated into parts, each of which is an exact part of some quantity, the cost of which is known, the costs of each of these may be found by division, and their sum will be the total cost. On these principles the Rules for the two cases in Practice are founded. Note to the observations in page 10 of the · Practice.' 1. It is axiomatical that every even number is divisible by 2. 2. Since any number of hundreds is divisible by 4, therefore any number is divisible by 4, in which the number composed of the last two digits is divisible by 4. Thus 1364 is divisible by 4, because 1300, and 64, are each so divisible. 3. Since any number of thousands is divisible by 8, therefore any number is divisible by 8, in which the number composed of the last 3 digits is divisible by 8. Thus 10512 is divisible by 8, because 10000, and 512, are each so divisible. 4. The truth of Obs. 4 is evident. 5. Any number of tens is divisible by 5, and 5 is divisible by 5, therefore any number ending with 5 is evidently divisible by 5. 6. Since 10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1, &c. therefore any number as 6564 may be put into this form : 6 X 999 +6 +5 X 99 +5 + 6 x 9 +6 + 4 or (6 X 999 + 5 X 99 + 6 x 9) + (6 +5+6+4) of which the first part is divisible by 3 and by 9; therefore, if the latter part be divisible also by 3 and by 9, the whole number will be so. 7. Since 10 = 11 – 1; 100 = 99 +1; 1000 = 1001 — 1; 10000 = 9999 + 1, &c. therefore any number as 57694 may be put into this form : S 6 5 X 9999 + 5+7 X 1001 — 7 + 6 X 99 + 6 +9 X 11 - 9+4 or (5 x 9999+7 1001 +6 X 99+9x11)+{(5+6+4) — (7+9)} of which the first part is divisible by 11 ; therefore if the latter part (which is the difference of the sums of the digits in the odd and even places) be also divisible by 11, the whole number will be so. Note to Prop. 4. It is important to observe, that all conclusions respecting abstract numbers are arrived at by means of reasoning upon concrete numbers. For, abstract numbers being symbols of an operation, viz.—repetition, we have no means of ascertaining the equivalence of two or more operations, except by considering their effect upon some concrete quantity. Thus because we find that 2 articles added to 3 articles produce 5 articles, we conclude, therefore, that the aggregate result of the operations denoted by 2 and by 3 is the same as that of the one operation denoted by 5, and consequently that the two operations 2 and 3 are together equivalent to the one 5. By this means it will be seen, that the Rule for Addition of numbers is obtained. For it being established, that the sum of several numbers of articles is the same as the sum of the smaller collections into which these may be formed, it is hence concluded that the operation, which is equivalent to the aggregate of several operations, is also equivalent to the aggregate of those to which the former are equivalent. The same remarks apply to the Rule of Subtraction. Note to Prop. 6. A product is defined to be the sum of several, the same numbers, and the operation, by which the product is obtained, is called Multiplication; the number, which is repeated, being the multiplicand, that, which shows how often it is repeated, being the multiplier. Hence it is evident that the multiplier is always an abstract number, but the multiplicand may be abstract or concrete. In the Prop. it is proved, that the repetition of three things 4 times is the same as the repetition of four things 3 times, and hence it is concluded that 4 times 3 are equal to 3 times 4, and that the abstract numbers in the multiplier and multiplicand may be interchanged. Note to Prop. 9. The Rule for Multiplication by a composite number might be deduced immediately from the nature of the product of two abstract numbers. Such a product is a number, representing an operation equivalent to the repetition of that denoted by the multiplicand, as often as there are units in the multiplier. Now an abstract number is the symbol of the operation of repetition, or multiplication; therefore multiplication by a composite number is equivalent to the repetition of the multiplication by one factor, a as often as there are units in the other; and the result may be obtained by successive multiplications by the factors. Note to Prop. 12. The first idea of Division is, that it is the operation of finding what is that number or quantity which, repeated a number of times equal to the divisor, will produce the dividend. Hence the divisor must be abstract, the dividend either abstract or concrete. When the dividend is abstract, we may put the definition into another form, and say that the object of Division is to find the symbol of the operation, which, repeated a number of times equal to the divisor, is equivalent to that denoted by the dividend. But since the repetition of the operation denoted by one number, as often as there are units in another, is equivalent to the successive performance of the operations denoted by the two numbers, therefore the object of Division might be defined to be, the finding the symbol of an operation, upon whose result if that denoted by the divisor be performed, a result will be obtained the same as by performing the operation denoted by the dividend. This is the view taken of Division in Prop. 35, 2nd case. Again, since 3 times 4 are equal to 4 times 3, therefore the symbol of the operation which, repeated 3 times, is equivalent to that devoted by 12, is the symbol of the number of times, which the operation denoted by 3 must be repeated to be equivalent to the same operation ; i.e. the third part of 12 is the number of times that 3 is contained in 12. Hence has arisen the ordinary definition of Division, as applied to abstract numbers; which is the view taken of it in Prop. 35, 3rd case. Note to Prop. 26. In the 6th conclusion read “which 2 units are of 5 units.” The 8th conclusion follows from Prop. 35, 3rd case. Note to Prop. 27. а Хc This Prop. may be proved otherwise, thus : 6 Хc operations of Multiplication by a X c, and Division by 6 Xc; or of Multiplication by a and by c, and Division by b and by c. Now the order in which these operations are performed is a matter of indifference; therefore having multiplied by a and by c, let us divide by c; we thus of course obtain the same result as by merely multiplying by a. We now have still to divide а Хc by b: so that the operations denoted by Xe are equivalent to those denoted by 6: Note to Prop. 29. The sum of a series of fractions, considered abstractedly, is the symbol of some operations, which are equivalent to those represented by the several a |