be compared with each other by reducing the fractions to a common denominator; then that will be the greater ratio which has the greater numerator. Ratios are commonly written by placing two points between the antecedent and consequent; thus a: 6 expresses the ratio of a to b, and is read a is to b. 99. Proposition 1st. A ratio of greater inequality is diminished by adding the same quantity to both its terms; whereas a ratio of lesser inequality is increased by adding the same quantity to each of its terms. atx For is a ratio of greater inequality, and if c be added atx+c to each of its terms, it becomes Reducing these ratios a+c a taxtac+cx to a common denominator, the first becomes a+as+ac ala+c) and the second which is evidently less than the a(a+c) a cx first by a(a+c) at a a Again, let be a ratio of lesser inequality; add c to a-*tc each of its terms, and it becomes ; reducing these a+c a* +ac-03-03 to a common denominator, they become and ax_axtac a(a+c) where the second is evidently greater than the first a(a+c) by ala+c) Q. E. D. 100. Proposition 2d. A ratio of greater inequality is increased, and a ratio of lesser inequality diminished, by subtracting the same quantity from each of its terms. atx Let be a ratio of greater inequality, take c from each ato of its terms, and it becomes ; reducing these to a a' tax-ac-C3 common denominator, the first becomes and aa tax-ac alac) the second which is greater than the former by ala--c) aa-c) be a ratio of lesser inequality ; take c from each of its terms, and it becomes ; reducing -ax-actor these to a common denominator, they become ala-c) aa_ax-ac and which is evidently less than the former a(a–c) Q. E. D. ala-c) a-C cx Again, let a a-C 0-C by 101. Prop. 3d. A ratio is not altered by multiplying or dividing its terms by the same quantity. Let a : 6 be any given ratio, then it is identical with c с с 5= or a mean PROPORTION. 102. The equality of two ratios constitutes a proportion; hus if a: 6 be equal to c:d, the two constitute a proporion, and are written thus; a:b::c:d, or a:b=c:d, and read, a is to b as c is to di consequently, since the ratio of a to b is D, di we have in which a and care called antecedents, and b and d consequents : also a and d are called extremes, and c and 3 means. Art. 14. 103. Prop. 1. In every proportion the product of the extremes is equal to the product of the means. For if a:b=c;d, then =å, multiplying both sides by bd we have ad=bc. Q. E. D. NOTE. If a:b=b:c, then b is called a mean proportional beween a and c, and c is called a third proportional to a and b; and by Prop. 1) it is evident that b2=ac; hence b=Vac, proportional between two quantities, is the square root of their product. Prop. 2. Two equal products can be converted into a proportion by making the factors of the one product the extremes, and the other the means. For if ad=bc, dividing both sides by bd, we have = ; hence a:b::c:d. Q. E. D. 104. Prop. 3. If four quantities be proportional, they are also proportional when taken inversely; that is, the second is to the first, as the fourth to the third. Since bd =>= and hence b:a=d:c. Q. E. D. 105. Prop. 4. If four quantities be proportional they are also proportional when taken alternately; that is, the first is to the third as the second is to the fourth. For 6 since = , if both sides be multiplied by and the common factors cancelled from the numerator and denomina с a с a d tor on both sides, we have .. a:c=:d. R. E. D. d 106. Prop. 5. When four quantities are proportional, they are also proportional by composition; that is, the sum of the first and second is to the second, as the sum of the third and fourth is to the fourth. Since = 6+1=+1; hence a+b_c+d d Therefore a+b:b=c+d:d. Q. E. D. 107. Prop. 6. When four quantities are proportional, they are also proportional by division; that is, the differ. ence of the first and second is to the second as the difference of the third and fourth is to the fourth. a-6 6Since = 6-1=-1; hence 6 Therefore a_b:b=(-d:d. Q. E. D. 108. Prop. 7. When four quantities are proportional, they are also proportional by mixing; that is, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference. a+b c d Q and Prop. 6th b d a+b ctd Х X hence 6 amb d Cad and therefore a+b:a_l=c+d:-d. Q. E. D. 109. Prop. 8. Quantities are proportional to their equimultiples .. Let a and b represent any quantities and ma and mb any equimultiples of them, then a:b=ma: mb. ; Q. E. D. mb Where m is any quantity, whole or fractional. 110. Prop. 9. The like powers and roots of proportional quantities are proportional. Since .. a" : 3"=ch:d". Q. E. D. dn Where n may be either whole or fractional, and consequently represent either a power or a root. 111. Prop. 10. If two proportions have the same antecedents, another proportion may be formed, having the consequents of the one for its antecedents, and the consequents of the other for its consequents. For if a : 6::c:d, and a:e::c:f, then 7 =à, and by inversion, =; a ma For a са" a e wherefore e: b=f:d, where e and f, the consequents of the one, are the antecedents, and b and d, the consequents of the other, are consequents. Q. E. D. 112. Prop. 11. If the consequents of one proportion be the antecedents in another, a third proportion will arise, having the same antecedents as the former, and the same consequents as the latter. Let a: b=c:d, and b:e=d:f; then a:e=c:f; for b hence ✓ ā ū multiplying these equals together, i x =áx. =; ; hence a: esc:f. Q. E. D. 113. Prop. 12. If there be any number of proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents. Let a:b=c:d=e:f=g:h; then a:b=a+c+e+g:6+d+f+h. For ab=ba, and Prop. Ist ad=bc, af=be, and ah=bg; herefore by adding equals to equals, we have ab +ad+af tah=ba + bc+be+bg; hence alb+d+f+h)=(a+c te+9); and therefore by Prop. 2d we obtain a: b=a+c+e+9:5+d+f+h. Q. E. D. 114. DEFINITION. When any number of quantities is n continued proportion, the first is said to have to the hird the duplicate ratio of the first to the second, and the irst is said to have to the fourth the triplicate ratio that he first has to the second. 115. Prop. 13. The duplicate ratio is the same as the atio of the squares of the terms expressing the simple ratio ; ind the triplicate ratio is the same as the ratio of the cubes of the terms expressing the simple ratio. Let a:b=b: :c=:d, then a: c=a: 62. And a : 03-a: 73. a b for since Х d lence a:c=a: 62. b с also X-X d XX and hence a : d=23: 13. R. E. D. a 116. Prop. 14. The product of the like terms of any numerical proportions are themselves proportional. For if a :b=c:d, then = с = m and b Х hence aei : bf k=cgl : dhm. Q. E. D. 117. Prop. 15. If there be three magnitudes, a, b, c, and other three, d, e, f, such that a : b=d; e, and 6: cre:f, then a :c=d:f. d 6 6 For and hence Х Х a d and therefore f consequently a : c=d:f. с Q. E. D. INTEREST. 118. INTEREST is the allowance given for the loan or forbearance of a sum of money, which is lent for, or becomes due at a certain time; this allowance being generally estimated at so much for the use of L.100 for a year. The money lent is called the principal, the sum paid for its use is called the interest, the sum of the principal and interest is called the amount, and the interest of L.100 for one year is called the rate per cent. Interest is either Simple or Compound. Simple interest is that which is allowed upon the original principal only, for the whole time or forbearance. 119. PROBLEM 1. To find the simple interest of any sum for any period, and at any given rate per cent. Let r=the interest of one pound for a year, p=the principal or sum lent, t=the time of the loan in the interest of the given principal for the given time, and a=the amount of the given principal and its interest for the time t; then we will obviously have the following relations among the quantities : 1: pt=r:i.: i=prt. (1.) and hence a=p+prt=P(1+rt). (2.) (3.) to (4.) 'rt pr years, ir a also p=1 tot a-P. pr |