EXPLANATION of THE CHARACTERS or MARKS USED IN THE FOLLOWING WORK +, Plus or more, the sign of addition ; as AD + DC, signifies that the line AD is to be increased by the line Dc ; and 4+3 signifies that the number 4 is to be increased by the number 3. -, Minus or less, the sign of subtraction, and shows that the second quantity is to be taken from the first; as CB-GB shows that the line CB is to be diminished by the line ge. x , Into or ly, the sign of multiplication; as Ed x DC signifies the rectangle formed by the lines ED and DC, and a × b expresses the product of the quantity a by the quantity b. Also arb or ab signifies the same thing. - - PB . . +, Divide by, as PB--cs, ori signifies that PB is to be divided by cs. AB", AB’, signify the square and the cube of AB; also Tolo signifies that 14 is to be involved to the third power, and then the fourth root is to be extracted. - 1 + w/A or A*, 3 y A or A*, express the square and cube root of A. =, Equal to, as AB = CD, shews that AB is equal to CD. ~, Difference, as A- B, shews that the difference between A and B is to be taken. A vinculum or parenthesis, serves to link two or more quantities together, as A+B x m, or (A+B). m., signifies that A and B are first to be added together, and then to be multiplied by the quantity m. { . . . . l Proportion, A : B : : C : d signifies that a has . . . . to B the same ratio which chas to D, and is to as is tos is usually read A is to B as c is to D. ... Therefore, z Angle, as Z. A. signifies the angle A. -a Greater than, as A-1B, shows that A is greater than B. r-Less than, as AE-B, shows A to be less than B. The other characters are explained among the definitions in the work. . . . . . N.B. The letters within the parentheses, at the beginning of the different paragraphs of the work, are for references. Thus, (C. 2.) refers to the article marked (C) at page 2. ; (H. 25.) refers to the article marked (H) at page 25, and so on. ERRATA. * . Page 93 and 94, in the note, for Chap. XI. read Chap. XIV. Page 305, line 10, for 5th of October, read 5th of August. (A) Deftnition. LOGARITHMS are a series of numbers contrived to facilitate arithmetical calculations; so that by them the work of multiplication is performed by addition, division by subtraction, involution by multiplication, and the extraction of roots by division. They may therefore be considered as indices to a series of numbers in geometrical progression, where the first term is an unit. Let 1. r". r". r". r". " .. r", &c. be such a series, increasing ... ', " : " ... ‘rs 1 ; which last series, agreeably to the established notation in algebra, may be thus expressed, 1 . r-1. r-2 . r-*. r–4 . r-5 . r-6, &c. Here the common ratio is r, and the indices 1. 2. 3, &c. or—1.—2.— 3, &c. are logarithms. Hence it is obvious, that if a series of numbers be in geometrical progression, their logarithms will constitute a series in arithmetical progresssion. And, where the series is increasing, the terms of the geometrical progression are obtained by multiplication, and those of the arithmetical progression, or logarithms, by addition; on the contrary, if the series be decreasing, the B terms of the geometrical progression are obtained by division, and those of the arithmetical progression by subtraction. The same observations apply to logarithms when they re 1 fractions, thus if rn denote any number, then will 1 . .”. 2 3 4 5 6 rn , rn . ros. r. ro, &c. constitute an increasing series of numbers in geometrical progression, of which the indices (B) Considering logarithms as indices to a series of numbers in geometrical progression, where r and n may represent any numbers whatever, it follows that there may be as many different kinds of logarithms as there can be taken different sorts of geometrical series: numbers in very different progressions may likewise have the same logarithms, and on the contrary, the same geometrical series may have different series of logarithms corresponding to them, but in every case the logarithm of 1 is 0. (C) The tables of logarithms in common use are constructed upon a supposition that r = 10; hence it appears that the logarithm of any number whatever is the index of some power of 10. - Thus, the logarithm of 10 is 1, being the index of 10'; the logarithm of 100 is 2, being the index of 10°; the logarithm of 1000 is 3, being the index of 10°; the logarithm of 10000 is 4, being the index of 10°, &c. Hence the logarithms of all numbers between 1 and 10 will be greater than 0 and less than 1, that is, they will be decimals; between 10 and 100 they will be greater than 1 and less than 2, that is, they will be expressed by 1 with decimals annexed; between 100 and 1000 the logarithms are expressed by 2 with decimals annexed; between 1000 and 10000 they are expressed by 3 with decimals annexed. - - 1 1 . - Again, the logarithm of I. Fiji is-1, being the index —l - 1. 1 . - of 10 ; the logarithm of Ty- I, is-2, being the index –2 1 1 . - of 10 ; the logarithm of Too Fio, is-3, being the index of 10 , &c. Hence the logarithms of all numbers decreasing from 1 to 1 will be expressed by— 1, with decimals annexed; decreasing from 1 to 01, they will be expressed by –2, with decimals annexed; decreasing from 01 to 001, by –3, with decimals annexed, &c. * - And, universally, if 10+ = a, then a is the logarithm of a . where a may be any number from 1 to 101,000, the extent of our best modern tables. If 10**=2, then r=0:30.103 the log. of 2. If 10+ =3, then r=0.47712 the log. of 3. If 10* =4, then r=0.60206 the log. of 4. . If 10° = 50, then r=1-69897 the log. of 50. If 10* =94261, then wit:497433 the log. of 94.261. The logarithm of 9,4261 = .97433 = or, viz. 10' = •97433 10 =9:4261, multiply by 10. 1-97 Then 10’’’=10” =94.261. Multiply again by 10, •97433 And 10” =10”=942.61. • Multiply a third time by 10, •97 Then 10” =10”=9426-1. Multiply once more by 10, multipying by 10, we divide by 10 in the equation 10r-9-4261, then will * The design of this chapter is to show the nature and properties of logarithms, and not the different methods of constructing them. |