will be enunciated, "&c.," "and so on," " continued according to the same law." PROPOSITION I. [AXIOM.] The whole is equal to the sum of all its parts. (1) This proposition is an axiom, that is, evident of itself; no words about it, therefore, can make it any plainer. (12) Corollary 1. The whole is greater than any of its parts, or, the whole exceeds any of its parts by those which are excepted-otherwise the whole would differ from the sum of all its parts. Cor. 2. Quantities which are equal in all their parts, are (13) said to be equal to each other; for the whole is known by its parts --or, quantities which are not evidently identical, can be compared only by a resolution into like or unlike parts. Cor. 3. Quantities which, however resolved, are unequal (14) in any of their parts, are not equal to each other (13). Cor. 4. Quantities which are equal to each other, are (15) equal in all their parts; for, if some of their parts were unequal, they would, by (14), be themselves unequal; .. Cor. 5. Unequal quantities are not equal in all their parts. (16) Cor. 6. Quantities which are equal to the same or equal (17) quantities, are equal to each other; for they are equal in all their parts, (1), (15). Cor. 7. Quantities measured by the same or equal quantities, (1.) are equal to each other; for equality of measures implies equality of parts, whether the measuring quantities be of the same kind with those measured or not. Thus, two masses of lead are equal in weight when they both contain the same number of pounds, or when they both contain the same number of cubic inches. Cor. 8. Quantities are to each other as their measures; .. (19) Cor. 9. Of quantities having unequal measures, that is (110) the greater to which the greater measure belongs. Cor. 10. If the same or equal quantities be increased or (111) diminished by the same or equal quantities, the resulting quantities will be equal to each other; since they will be equal in all their parts, (15), (1). Cor. 11. If the same or equal quantities be multiplied or (112) divided by the same or equal quantities, the resulting quantities will be equal; since multiplication is repeated addition, and division a continued subtraction. Cor. 12. If equal quantities be raised to the same pow- (113) ers, or the same roots be taken of them, the resulting quantities will be equal; since a power is formed by continued multiplication, and a root is extracted by the converse operation. Cor. 13. The same or equal quantities, by the same or (11) equivalent operations, give the same or equal quantities. Cor. 14. Quantities satisfying the same or equivalent (116) conditions, are equal to each other. Cor. 15. Unequal quantities, by the same or equivalent (116) operations, will continue to be unequal, and in the same sensethat is, the greater will be the greater still (114). See also (111), (112), (118). Thus, if a be greater than b,[a> b], a increased by c will be greater than b increased by c [a+c,>b+c], a diminished by c will be greater than b diminished by c,[a—c > b—c], m times a will be greater than m times b,[ma > mb], &c. Cor. 16. If inequalities, taken in the same sense, be (117) added, the result will be an inequality also in the same sense (116). Thus, if a be> b and c > d, then a +c>b+d. Cor. 17. When inequalities, whose differences are the same, (118) are added in a contrary sense, the result will be an equality (13). Thus, if a be as much > b as c is <d, then a + c will = b+d. Cor. 18. When inequalities are added in a contrary sense, (119) the sense of the resulting inequality will be that of the greater (118). Thus, if a> b and c <d, then will a+c>b+d, provided the difference between a and b be greater than the difference between c and d,[a−b>d - c, .. a > d − c + b. a+c>b+d]. Cor. 19. If inequalities, taken in the same sense, be sub- (120) tracted, the one from the other, the resulting inequality will be in the same or a contrary sense, according as the minuend is the greater or less inequality. PROPOSITION II. [COROLLARY FROM 18.] Magnitudes which may be made to coincide throughout, (2) are equal to each other. The magnitudes are equal in all their parts. Cor. 1. When one magnitude embraces another without (2) being filled by it, the first is greater than the second (1). Cor. 2. Magnitudes measured by the same or equal mag- (23) nitudes, are equal to each other (18). Cor. 3. Magnitudes are to each other as their measures (1,). (24) Cor. 4. Of magnitudes having unequal measures, that (25) possessing the greater measure is the greater. Scholium I. It is sometimes convenient to make a distinction between equal and equivalent, but the terms will generally be used as synonymous. Scholium II. It is obvious that all propositions requiring demonstration, must be founded, either directly or indirectly, upon those which do not, or on axioms; and hence our first proposition becomes the source of a vast amount of knowledge. Def. 4. A coëfficient is a figure employed to show how many times a letter is taken; thus, in 3a, 3 is the coëfficient of a, and 3a = a +a+a. A letter may be regarded as a coëfficient, as n in na = a+a+a+... [n times]. Def. 5. Operations are said to be relatively free when the result is the same in whatever order they are executed, the one after the other. Thus, O, O2, the two parts of a compound operation, are relatively free when Additions, subtractions, additions and subtractions, are (3) relatively free operations—that is, the terms of a polynomial may be inverted at pleasure. 1o. Additions. That 4+3 is to 3+4 will be evident from counting the units into which the two sums are resolvable; thus and '. (13) 4+3=(1+1+1+1)+(1+1+1), 3+4=(1+1+1)+(1+1+1+1) ; 4+3=3+4. So a+b=(1+1+1 + ... [a units]) + (1 +1 +1 + ... [b units]) =1+1+1+... [a+b] = (1 + 1 +1 + ... [b] ) + (1 +1 +1 + .....[a]) = b+a, which was to be proved. 20. Subtractions. The remainder arising from diminishing a units first by b units and. then by c units will be found the same as that obtained by diminishing a first by c and then by b, or a − b — c = a -c-b; for, let a contain r+b+c units, or a=r+b+c, which (1°) =r+c+b; then (111) abr+c, subtracting b from both sides, and or and a − b — c = r, subtracting c from the last equation— a c-br, subtracting b from the last; .. (17) a b-c=r=a-c-b, Q. E. D.* 3°. Additions and subtractions. a+b-ca-c+b; for we obviously have the same number of units whether we diminish the a+b units by taking the c units from b or from a. Q. E. D. PROPOSITION IV. Multiplications, divisions, multiplications and divisions, (4) are relatively free operations. = 10. Multiplications. We may know that 3 4 4.3 by resolving the numbers into their component units, and setting these down in an orderly way to count;, thus, by counting, we find 4 units repeated 3 times, the same as 3 units repeated 4 times, or 3 • 4=4.3. 3 units. 4 units. 1 1 1 1 a units. So, a units (= 1+1 +1 + ... [a]) repeated b times, is the same as b units (= 1+1 +1 + ... [b] ), repeat hence (112) a times b units, repeated c times will be equal to b times a units repeated c times, but b times a units, repeated c times, by what has just been demonstrated, is to c units repeated b times a times-and in order to repeat c units b times a times, we may multiply first by b and then by a; for, multiplying by b instead of ba, is multiplying by a number a times too small, and consequently, the product, being a times too small, will be corrected by multiplying again by a; all which may be set down in symbols thus ; * "Quod erat demonstrandum," which was to be proved. where it will be observed that a is made to occupy every place in the product, and in like manner the same may be shown of b and c -and the reasoning may be extended by introducing additional factors at pleasure. Q. E. D, for 10. Cor. 1. The factors of a product may be grouped in multi- (4) plication at pleasure. Cor. 2. Any factor may be regarded as the coefficient to (43) the remaining factors of the product (42). 20. Division. Assuming any quantity, Q, to be divisible by others, as b and c, is obviously the same as assuming Q to be resolvable into factors, two of which are b and c; hence, Q being divisible by b and c, denoting the third factor by a, we have .. (112) Q = abc; Q: cab, dividing both sides by c, and observing that the product abc is divided by c by omitting the multiplier c; .. dividing the last by b, there results But (10) and .. (17) Q: b = ac, (Q: b): c = a ; (Q b) c = (Q c) b. Q. E. D. for 2°. 3°. Multiplication and Division.—Let Q be divisible by b, or Q = ab; and, multiplying by c, (Q: b) again hence = a, and the proposition is proved. PROPOSITION V. Additions and Subtractions in regard to multiplications (5) and divisions, are relatively free. 1°. Quantities otherwise alike are added and subtracted by (52) adding and subtracting their coëfficients. |