in Theor. 4. would in a scale of eight belong to seven, and those in Theor. 5. to nine. If twelve was the root of the scale, the former properties would belong to eleven, and the latter to thirteen. A LGEBRA may be employed in expressing the relations of magnitude in general, and in reasoning with regard to them. It may be used in deducing not only the relations of number, but aso those of extension, and hence those of every species of quantity expressible by numbers or extended magnitudes. In this Appendix are mentioned some examples of its application to other parts of mathematics, to physics, and to the practical calculations of business. The principles and suppositions peculiar to these subjects, which are necessary in directing both the algebraical operations, and the conclusions to be drawn from them, are here assumed as juft and proper I. I. Algebra has been successfully applied to almost every branch of mathematics; and the principles of these branches are often , advantageously introduced into algebraical calculations. The application of it to geometry has been the source of great improvements in both these sciences. On account of its extent and importance, it is here omitted, and the principles of it are more particularly explained in the third part of these elements. In this place shall be given an example of the use of logarithms in resolving algebraical questions. Note. When logarithms are used, let (1.) denote the logarithm of any quantity before which it is placed. Ex. To find the number of terms of a geome trical series, of which the sum is 511, the first term 1, and the common ratio 2. From Sect. 2. Chap. 6. it appears that arn and in this problem, s, r, and a -a I are १ a are given, and n is to be found. By redu sxr-ita, cing the equation, guna and from the known property of logarithms n Xlir= los xp 1+a-la, and n= 1.5 xr--ita-la Ir =9 But here s=511, a=1,,r=2, and n= 0.3010300 An equation of the following quadratic form a2x+ 2ba*=£c may be resolved by logarithms. ist, By scholium of chap. 5. 2;=+b/Ec. And then X is discovered in the same manner as in the preceding example. Thus, let a=2, b=10, and c=96, and the equation 22--20 X 2* =-96. ift, 2*=10+14=12 or 8. If 1.8. 2*=8, then x=12.=3, and 20—20X 2 = 96 is a true equation. If 2* =12, then 1.12._1.0-91812 -1.2 ber 0.3010300=3.5849, and this num ber being inserted for x in the given equation, by means of logarithms, will answer the conditions. Ex. 3. The sum of 2000 l. has been out at interest for a certain time, and 500 1. has been at intereft double of that time, the whole arrear now due, reckoning 4 per cent. compound interest, is 6000l. What were the times ? By the rules in the third part appendix for compound intereit, it is plain that if R=1,04, and the time at which the 2000 l. is at interest be x, the arrear of it will be 2000 XR*. The arrear of the 500l. is 500 x R2*, hence 500 x R2*+2000 XRx=6000. This resolved gives R*= 2, 1.2. andx=iR=17.67 +, nearly, that is, 17 years and 8 months nearly, and the double is 35 years and 4 months; which answer the conditions. 1 II. Application of Algebra to Physics. Physical quantities which can be divided into parts that have proportions to each other, the same as the proportions of lines to |