Base = √(hypoth. + perpend.) × (hypoth. - perpend.)
Perpendicular = √(hypoth. + base) × (hypoth. - base.)
TRIGONOMETRY, WITHOUT LOGARITHMS.[49]
“In all the more elaborate, and refined operations of trigonometry, it is not only desirable, but necessary to employ some of the larger logarithmic tables, both to save time, and to ensure the requisite accuracy in the results. But in the more ordinary operations, as in those of common surveying, ascertaining inaccessible heights, and distances, reconnoitring, &c., where it is not very usual to measure a distance nearer than within about its thousandth part, or to ascertain an angle nearer than within two or three minutes, it is quite a useless labour to aim at greater accuracy in a numerical result. Why compute the length of a line to the fourth, or fifth place of decimals, when it must depend upon another line, whose accuracy cannot be ensured beyond the unit’s place? Or, why compute an angle to seconds, when the instrument employed does not ensure the angles in the data beyond the nearest minute? In the following Table are brought together the natural sines, and cosines, to every degree in the quadrant, and this table will be found sufficiently extensive, and correct for the various practical purposes above alluded to. The requisite proportions must, it is true, be worked by multiplication, and division, instead of by logarithms. Yet this by no means involves such a disadvantage as might seem, at first sight. For when the measured lines are expressed by three, or at most, four figures, the multiplications, and divisions are performed nearly as quick, and in some cases quicker, than by logarithms. Then as to accuracy, even in cases where the computer will have to take proportional parts for the minutes of a degree, the result may usually, if not always, be relied upon to within about a minute.”
TRIGONOMETRIC RATIOS.
Natural sines, and cosines to every degree in the quadrant, radius being 1·000000.
| Deg. | Sines. | Cosines. | Deg. | Sines. | Cosines. | ||
| 0 | ·00000 | 1·00000 | 90 | ||||
| 1 | ·01745 | ·99985 | 89 | 26 | ·43837 | ·89879 | 64 |
| 2 | ·03490 | ·99939 | 88 | 27 | ·45399 | ·89101 | 63 |
| 3 | ·05234 | ·99863 | 87 | 28 | ·46947 | ·88295 | 62 |
| 4 | ·06976 | ·99756 | 86 | 29 | ·48481 | ·87462 | 61 |
| 5 | ·08716 | ·99619 | 85 | 30 | ·50000 | ·86603 | 60 |
| 6 | ·10453 | ·99452 | 84 | 31 | ·51504 | ·85717 | 59 |
| 7 | ·12187 | ·99255 | 83 | 32 | ·52992 | ·84805 | 58 |
| 8 | ·13917 | ·99027 | 82 | 33 | ·54464 | ·83867 | 57 |
| 9 | ·15643 | ·98769 | 81 | 34 | ·55919 | ·82904 | 56 |
| 10 | ·17365 | ·98481 | 80 | 35 | ·57358 | ·81915 | 55 |
| 11 | ·19081 | ·98163 | 79 | 36 | ·58778 | ·80902 | 54 |
| 12 | ·20791 | ·97815 | 78 | 37 | ·60181 | ·79863 | 53 |
| 13 | ·22495 | ·97437 | 77 | 38 | ·61566 | ·78801 | 52 |
| 14 | ·24192 | ·97030 | 76 | 39 | ·62932 | ·77715 | 51 |
| 15 | ·25882 | ·96593 | 75 | 40 | ·64279 | ·76604 | 50 |
| 16 | ·27564 | ·96126 | 74 | 41 | ·65606 | ·75471 | 49 |
| 17 | ·29237 | ·95630 | 73 | 42 | ·66913 | ·74314 | 48 |
| 18 | ·30902 | ·95106 | 72 | 43 | ·68200 | ·73135 | 47 |
| 19 | ·32557 | ·94552 | 71 | 44 | ·69466 | ·71934 | 46 |
| 20 | ·34202 | ·93969 | 70 | 45 | ·70711 | ·70711 | 45 |
| 21 | ·35837 | ·93358 | 69 | ||||
| 22 | ·37461 | ·92718 | 68 | ||||
| 23 | ·39073 | ·92050 | 67 | ||||
| 24 | ·40674 | ·91355 | 66 | ||||
| 25 | ·42262 | ·90631 | 65 | ||||
| Cosines. | Sines. | Deg. | Cosines. | Sines. | Deg. | ||
“The preceding table is so arranged that for angles not exceeding 45 degrees, the sine, and cosine for any number of degrees will be found opposite to the proposed number in the left hand column, and in the column under the appropriate word. When the number of degrees in the arc, or angle, exceeds 45 degrees, that number must be found in the right hand column, and opposite to it in the column indicated by the appropriate word at the bottom of the table. Thus, the sine, and cosine of 36 degrees are ·58778 and ·80902 respectively, the radius of the table being unity, or 1. The taking of proportional parts for minutes can only be done correctly in those parts of the table where the differences between the successive sines, &c., run pretty uniformly. Suppose we want the natural sine of 20° 16′. The sine of 21 degrees is ·35837, that of 20 degrees is ·34202; their difference is ·1635. This divided by 60 gives 27·25 for the proportional part due to 1 minute, and that again multiplied by 16 gives 436 for the proportional part for 16 minutes. Hence the sum of ·34202 and 436, or ·34638, is very nearly the sine of 20° 16′. But the operation may often be contracted by recollecting that 10 minutes are ⅙, 15 minutes are ¼, 40 minutes are ⅔ of a degree, and so on. Observe, also, that for cosines the results of the operations for proportional parts are to be deducted from the value of the required trigonometrical quantity in the preceding degree.”
APPLICATION OF TRIGONOMETRY, WITHOUT LOGARITHMS, to the determination of Heights, and Distances.