The method was originated by Col. Jared Mansfield, whose great acquirements in science introduced him to the notice of President Jefferson, by whom he was appointed surveyorgeneral of the North-Western Territory. May it be permitted to one of his pupils, and a graduate of the Military Academy, further to add, that at the organization of the institution in 1812, he was appointed Professor of Natural and Experimental Philosophy. This situation he filled for sixteen years, when he withdrew from the academy to spend the evening of his life in retirement and study. His pupils, who had listened to his instructions with delight, who honored his learning and wisdom, and had been brought near to him by his kind and simple manners, have placed his portrait in the public library, that the institution might possess an enduring memorial of one of its brightest ornaments and distinguished benefactors. Military Academy, West Point, 1835. INTRODUCTION. CHAPTER I. Of Logarithms. 1. The nature and properties of the logarithms in common use, will be readily understood, by considering attentively the different powers of the number 10. They are, 10°=1 101=10 102=100 103=1000 101=10000 105=100000 &c. &c. It is plain, that the indices or exponents 0, 1, 2, 3, 4, 5, &c. form an arithmetical series of which the common difference is 1; and that the numbers 1, 10, 100, 1000, 10000, 100000, &c. form a geometrical series of which the common ratio is 10. The number 10, is called the base of the system of logarithms; and the indices, 0, 1, 2, 3, 4, 5, &c., are the logarithms of the numbers which are produced by raising 10 to the powers denoted by those indices. 2. Let a denote the base of the system of logarithms, m any exponent, and M the corresponding number: we shall then have, a" =M in which m is the logarithm of M. If we take a second exponent n, and let N denote the corresponding number, we shall have, a"=N in which n is the logarithm of N. If now, we multiply the first of these equations by the second, member by member, we have but since a is the base of the system, m+n is the logarithm MxN; hence, The sum of the logarithms of any two numbers is equal to the logarithm of their product. Therefore, the addition of logarithms corresponds to the mul tiplication of their numbers. 3. If we divide the equations by each other, member by member, we have, am M an - ; but since a is the base of the system, m-n is the logarithm of hence: M N If one number be divided by another, the logarithm of the quotient will be equal to the logarithm of the dividend diminished by that of the divisor. Therefore, the subtraction of logarithms corresponds to the division of their numbers, 4. Let us examine further the equations 10°=1 10' 10 102=100 103=1000 &c. &c. It is plain that the logarithm of 1 is 0, and that the logarithms of all the numbers between 1 and 10, are greater than o and less than 1. They are generally expressed by decimal fractions thus, log 2=0.301030. The logarithms of all numbers greater than 10 and less than 100, are greater than 1 and less than 2, and are generally expressed by 1 and a decimal fraction: thus, log 50 1.698970. The logarithms of numbers greater than 100 and less than 1000, are greater than 2 and less than 3, and are generally expressed by uniting 2 with a decimal fraction; thus, log 126 2.100371. The part of the logarithm which stands on the left of the |