Log.6 2.302585÷1.791759=.778151 Log. 10. 2.302585÷2.302585=1.000000 ́; and, Art. 37. Log. 4.301030+.301030=.602060 66 66 66 Log. 9 .477121+.477121.954243 Log. 10.301030+.698970=1.000000 Note Of this method, we add, that for those who may wish to compute an entire table, it may possess very slight advantages over that given by Day. But a rule for such an object will, it is believed, never be called for. In the treatise under consideration, the student is never required to do more than to calculate the logarithms of the prime numbers 2, 3 and 7; and in order to do this, by Day's rule, it is necessary to find the reciprocal of the base of Napier's system, which is of itself a considerable labor, and is rendered unnecessary by the other rule. But as it is not so much our object to give new rules, as to furnish solutions according to those which are given, we shall proceed first, to find the reciprocal of 2.302 585 092 944 045 668, and then to multiply this reciprocal by the logarithms of 2, 3 and 7, according to Napier's system, which will give the logarithms of these numbers according to Briggs' system. The division of 1 by 2.302 585 092 994 045 668, is so tedious, that it is desirable to have some method of detecting errors as they may occur in any part of the process. We shall, therefore, multiply the divisor by each of the numbers 2, 3, 4, 5, 6, 7, 8, 9; then during the process of division, when the divisor is multiplied into any one of these figures, its product may be compared with the previous product of the divisor into the same number. In cases where there is entire certainty of the accuracy of the first products, these may be used without a second multiplication. 2×2.302 585 092 994 045 668 3×2.302 585 092 994 045 668 4×2.302 585 092 994 045 668 4.605 170 185 988 091 336 6.907 755 278 982 137 004 9.210 340 371 976 182 672 5×2.302 585 092 994 045 668-11.512 925 464 970 228 340 6×2.302 585 092 994 045 668-13.815 510 557 964 274 008 7x2.302 585 092 994 045 668-16.118 095 650 958 319 676 8×2.302 585 092 994 045 668=18.420 680 743 952 365 344 8×2.802 585 092 994 045 668-20.723 265 836 946 411 012 Divisor, 2.302 585 092 994 045 668).434 294 481 903 251. 830, Quotient. 1.000 000 000 000 000 000 000 000 000 000 000 000 921 034 037 197 618 267 2 78 965 962 802 381 732 80 9 888 410 012 560 362 760 678 069 640 584 180 088 0 4 382 399 293 519 687 360 7 487 616 831 000 590 800 579 861 552 018 453 796 0 4 215 278 769 942 379 000 1 548 049 763 275 427 560 From this tedious process, we find the reciprocal of Napier's base to be .434 294 481 903 251 830. For ordinary purposes, and for the use which we shall make of it, it is not necessary that it should be carried to so great a number of decimals. As it is, it will answer for finding logarithms to almost any degree of accuracy, and serve equally well for finding them to only 6 or 7 figures. We now give, as before premised, the logarithms of the prime numbers 2, 3 and 7. VARIOUS EXPLANATIONS FROm art. 71, to art. 134. Corollary 2, Art. 96. The cosine of 30°=√3. Radius, cosine and sine constitute a right angled triangle, radius being the hypothenuse: Therefore, Euclid I. 47, Ra =cos2+sin2, and as cos 30°=sin 60°, if radius=1, the above equation becomes by transposing, cosa 30°-R2-sin3 30° 1. Extracting the square root of cos2 30° and The sine of 45°-the cosine of 45°, each being equal to radius, therefore R2=sin2 45°+cos2 45°-2sin2 45°. As R2=1, sin2 45°, and sin 45°=√: But the root of a fraction is equal to the root of the numerator divided by the root of the denominator, and in √, the numerator being 1, any root of it is 1; that is, √√1÷√2=1÷2= Four examples under article 108. 60" .0003373::18′′; x Multiplying extremes and means, 60x=.0060714 Dividing by 60, 1 x=. rection to be added to 9.6536631, the tangent of 24° 15', which added, becomes 9.65376429, the answer. Required to cotangent of 31° 50' 5". The cotangent of 31° 51'=10.2067440 = 60": .0002819::5": x Multiplying extremes and means, 60x=.0014050 10.2070259+.00002349=10.20700241, the cotangent of 31° 50' 5". Required the sine of 58° 14' 32". The sine of 58° 15' 9.9295989 The sine of 58° 14'-9.9295207 60".0000782::32′′ : x Multiplying extremes and means, 60x=.0000782 × 32, and x.00004004, which added to 9.9295207, becomes 9.92956074, the sine of 58° 14′ 32′′. Multiplying extremes and means, 60x=.0037216, and x= 00007869, which added to 9.7567815, becomes 9.75670281, the cosine of 55° 10' 26". Four examples under article 110. Required the angle belonging to the sine of 9.20621. Sin next greater 9° 16′ 9.2069059 Sin next less 9° 15' Difference, 9.2061309 .0007750 9.20621 9.2061309 .0000791, there fore by the rule, .0007750.0000791::60" x Multiplying extremes and means, .0007750x=.000047460", and x=6": therefore, the angle belonging to the sine 9.20621 is 9° 15' 6". : x. Multiplying extremes and means, .0000376 x = .0012360", and x=30". Therefore the angle belonging to the cosine 9.98157, is 16° 34′ 30′′. Required the angle belonging to the tangent 10.43434. 60": x. 69° 48' 10.4342367 10.4342367 .0003900: .0001033:: Multiplying extremes and means, .0003900x=.00061980, and x 16" Therefore the angle belonging to the tangent 10.43434, is 69° 48′ 16′′. Required the angle belonging to the cotangent of 10.33554. Cot next greater 24° 48′ 10.3356289 10.33554 |