The loga rithms of bers in crease very nearly in the same numbers In the first column are placed, underneath the letter N, the first four digits of the number, the fifth being written in the same line with it at the head of the successive columns. In the column headed 0, are written the mantissa of the logarithms of 26500, 26510, 26520,......, the three first digits being suppressed as long as they remain the same with those in the first line in the column headed 1, are written the four last digits of the mantissa of the logarithms of 26511, 26521, 26531, : in the column headed 2, the four last digits of the mantissæ of the logarithms of 26512, 26522, 26532,......: and so on for the remaining columns headed by the remaining nine digits, in their order. It will be observed that the mantissa of the logarithm of 26546 is 4239991 and that of the logarithm of 26547 is 4240154: this change of the third digit of the mantissa from 3 to 4, is indicated, in the place where it first occurs, by writing the four last digits thus 0154. 869. If the mantissa of the logarithms of the successive large num- numbers, which are given in the Extract from the tables in the last Article, be subtracted from each other, their difference will appear to be between .0000164 and .0000163, and will be proportion found, by actual reference of the Tables, to continue nearly the with the same for the whole series of numbers between 26500 and 26800: themselves. We are authorized to conclude, therefore, that the mantissæ of large numbers will increase, within small limits, very nearly in the same proportion with the numbers themselves, and that, consequently, if the mantissæ of the logarithms of two successive large numbers, and therefore their difference, be given, the mantissa of the logarithms of all intermediate numbers may be found approximately, if not accurately, by a simple proportion and conversely*: this property of logarithms is very important, as furnishing us with very simple and expeditious methods of greatly extending the range of the tables. To find the logarithm 870. Thus, if it was required to find the logarithm of a of a num- number of more than 5 digits (and therefore beyond the range ber of more of the ordinary tables), we should proceed as follows. than five places. We should find from the tables, the logarithm of the number This proposition will be easily shewn to be an immediate consequence of the logarithmic series, which will be given in Chapter xxxv. formed by the first five digits; and to this we should add, from the column of proportional parts headed Prop., the number 1 corresponding to the 6th digit andth of the number corresponding to the 7th digit, by advancing the number given in the column one place to the right: their sum would be the logarithm required to seven places of figures: and if the number of decimal places of the logarithms recorded in the Tables was greater than what is commonly given, the same method might be extended to find the logarithms of numbers of more than seven places. Thus, if it was required to find the logarithm of 265.4678, we Example. find from the tables number 871. Again, if it was required to find the number corre- To find the sponding to a non tabulated logarithm, we should proceed as correspondfollows: ing to a logarithm tables. We find the difference between the mantissa of the given not in the logarithm and the next inferior mantissa in the tables and we write down the corresponding number: in the column of proportional parts, we find the number which is equal to, or next below, the difference thus found and the digit opposite to it is the 6th digit of the number sought for: if there be a remainder, we subjoin a zero to it, forming a new difference (removed one place to the right) and the digit in the column of proportional parts which is opposite to the number equal to or next below, it, is the 7th digit of the number required; and similarly, if more digits than 7 are required to be determined. For if D be the difference in the mantissa corresponding to a unit of the D tabular number, and will be the differences, in conformity with the 10 D 100 property assumed in the text, corresponding to the two next inferior units in the 6th and 7th places respectively, and therefore and b D ferences, if a and b be the digits in those places: the first is found in the column of proportional parts opposite the digit a: the second, removed one place to the right, is found in the same column opposite the digit b. Example. Tables of logarithmic sines, cosines, tangents, &c. The reason why the logarithms of the nu merical values of the sines, cosines, &c. are in creased by 10. Thus, let it be required to find the number, whose logarithm is 1.233678, and whose precise value is not found in the Tables. 872. Inasmuch as the sines, cosines, tangents, secants, &c. of angles enter into formula which are the subjects of calculation equally with other symbols possessing assigned numerical values, a register of the logarithms of their successive values becomes equally necessary with that of the logarithms of the series of natural numbers. Tables or canons of natural sines, (Chap. XXVIII.) cosines, tangents, cotangents, secants, cosecants, will contain their successive values for every minute (and in some tables for every ten seconds) of all angles between 1′ and 45° and consequently between 1' and 90°, if taken in an inverse order, when the sine is replaced by the cosine, the tangent by the cotangent, and the secant by the cosecant: whilst tables of the logarithms of sines, cosines, tangents and cotangents, secants and cosecants, will contain the logarithms of their natural values increased by the number 10, arranged in precisely the same order, each page of the natural values being opposite to a page of their corresponding logarithmic values. 873. A very little consideration will shew the great convenience, for the purposes of calculation, of increasing the logarithms of the goniometrical quantities, as recorded in the tables, by the number 10: for the natural values of the sines and cosines are included between 0 and 1 and the characteristics of their logarithms are therefore necessarily negative: thus sin l' = .0002909: its unaltered logarithm is 4.4637261, If such logarithms, therefore, were registered in tables, their haracteristics and mantissa would have different signs and great confusion might thus be occasioned in the calculation, by means of them, of the values of formulæ in which such quantities occurred, particularly in the hands of calculators, as is very commonly the case, who have no very accurate knowledge of the principles of Algebra: it is for this reason, that the logarithms of all goniometrical quantities are increased by 10, and the logarithms of sin 1', sin 1o, sin 50°, cos 30', cos 30°, cos 85°, present themselves, therefore, in the tables, with the following values: 6.4637261, 8.2418553, 9.8842540, 9.9999836, 9.9375306, 8.9402960. tabulated of sines, and con 874. The number 10 is the logarithm of 10" and the regis- Transition tered or tabulated logarithms of the sines, cosines, and other gonio- from the metrical quantities are the proper logarithms of the products of logarithms those quantities and 1010; or if we adopt the ancient definitions, cosines,&c. we should say that the tabulated logarithms were those of the sines, to their natural cosines, &c. in a circle whose radius was 100 or 10000000000 logarithms (Art. 754, Note): but in the adaptation of formula to logarithmic computation, we may always pass from the tabulated to the proper logarithms of such quantities, by subtracting 10 from their characteristics and conversely: or we may follow the practice, which is generally most expeditious and convenient, of adding to or subtracting from, the final logarithm which results, 10 or any multiple of 10, which may obviously have been introduced by the use of tabulated instead of proper logarithms. The student will find, in the two following chapters, several examples not merely of the adaptation of expressions, involving goniometrical quantities to logarithmic computation, but likewise of the calculation of their numerical values by means of them. CHAPTER XXXIII. ON THE DETERMINATION OF THE SIDES AND ANGLES OF Equations press the 875. We have already shewn (Art. 833), that if a, b, c be the sides of a triangle, and A, B, C the angles opposite to them, general the following equations will express their general relations to each other, assuming c to be the primitive line. relations of the sides These equations may be derived immediately from the triangle, by the aid of the definitions of the sine and cosine: for AD AC cos 4 = b cos A, = b which is the first equation. And again, since CD = b sin A = a sin B, we get If the triangle be oblique-angled, as in Fig. 2, and if CD be drawn perpendicular upon AB produced, then the angle C or |