in Theor. 4. would in a scale of eight belong to Seven, and thofe 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. I APPEN APPENDIX то A PARTI. LGEBRA may be employed in expreffing 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 also those of extenfion, and hence those of every species of quantity expreffible by numbers or extended magnitudes. In this Appendix are mentioned fome examples of its application to other parts of mathematics, to phyfics, and to the practical calculations of business. The principles and fuppofitions peculiar to these subjects, which are neceffary in directing both the algebraical operations, and the conclufions to be drawn from them, are here affumed as just and proper. I. Algebra has been fuccessfully applied to almoft 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 fource 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 ufe of logarithms in refolving algebraical questions. Note. When logarithms are used, let (7.) denote the logarithm of any quantity before which it is placed. Ex. To find the number of terms of a geometrical feries, of which the fum is 511, the firft term 1, and the common ratio 2. From Sect. 2. Chap. 6. it appears that and in this problem, s, r, and a are are given, and n is to be found. By redu the known property of logarithms nXlr= l.sxr-1+a-l.a, and n= l.sxr-1+a-l.a Ir But here 511, a=1,、r=2, and n= In like manner, may any fuch equation be refolved, when the only unknown quantity is an exponent, and when it is the exponent only of one quantity. Ex. 2. An equation of the following quadratic form a2±2ba*=±c may be refolved by logarithms. Ift, By fcholium of chap. 5. a±b±√b ±c. And then 'x is difcovered in the fame manner as in the preceding example. Thus, let a=2,b=10, and c=96, and the equation 22-20X 2* =-96. 1st, 2*=10±√4=12 or 8. If 1.8. 3 2*8, then x= 1.2.3, and 26-20X23= -96 is a true equation. If 2*12, then 1.12. 1.0791812 -1.2 = 0.3010300-3.5849, and this num ber ber being inferted for x in the given equa tion, by means of logarithms, will answer the conditions. Ex. 3. The fum of 2000 1. has been out at intereft for a certain time, and 500 I. has been at intereft double of that time, the whole arrear now due, reckoning 4 per cent. compound intereft, is 6000l. What were the times? By the rules in the third part of this appendix for compound intereft, it is plain that if R1,04, and the time at which the 2000 l. is at intereft be x, the arrear of it will be 2000 XR*. The arrear of the 500 l. is 500 × R2, hence 500 × R2 +2000 XR*=6000. This refolved gives R*—2, 1.2. and x= 17.67+, nearly, that is, 17 L.R years and 8 months nearly, and the double is 35 years and 4 months; which answer the conditions. II. Application of Algebra to Phyfics. Phyfical quantities which can be divided into parts that have proportions to each other, the fame as the proportions of lines to |