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PORTABLE TRIGONOMETRY, WITHOUT

LOGARITHMS.

In all the more elaborate and refined operations of trigonometry, as well as their applications to navigation, astronomy, and geodesic surveying, it is not only desirable, but necessary, to employ some of the larger logarithmic tables, as those of Hutton, Babbage, Callet, Taylor, &c., 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 2 or 3, or in very rare cases within 1 minute, it is quite a useless labour to aim at greater accuracy in a numerical result. Why, for example, should I compute the length of a line to the 4th or 5th place of decimals, when it must depend upon another line, whose accuracy I cannot ensure 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?

Hence several mathematicians, as Euler, Legendre, Hutton, Bonnycastle, &c., have investigated approximating series and other rules, for solving the cases of trigonometry without tables; yet, however ingenious their researches may have been, they have not led to any results of practical value, but simply furnish so many proofs how easy it is for scientific men, in their investigations, to miss the point of real utility.

It is truly extraordinary, that amid all this search for expedients, the obvious method which I now beg to recommend and exemplify, has never been thought of. In the table which precedes this article, and which, set up with a bold, clear type, occupies only an octavo page, I have brought together the natural sines, tangents, and secants, to every degree in the quadrant; and have no doubt that this table, though only carried to five places of decimals, will be found

sufficiently extensive and sufficiently correct for the various practical pur poses to which I have adverted. And thus, the surveyor, the architect, the civil or military engineer, furnished with this table, in one page, with a box and 100 feet tape, a pocket sextant, or Schmalcalder's elegant portable theodolite, may take every angle and perform every computation that can occur in the most useful cases.

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 by four figures, (and to give more, only presents an appearance of accuracy which does not exist) the multiplications and divisions are performed' nearly as quick, and in some cases quicker (as will be seen) than by logarithms. Besides which, the operations may often be shortened, by resolving numbers into their component factors, and by other contractions well known to practical men. Nay, if this were not the case, the circumstance would not present any serious objection: for, when a computation is not to be performed once in a month, it does not greatly signify whether you complete it in ten minutes or in twenty.

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: and, recollecting that in the out-of-door operations he has it commonly at his option to fix his instrument at angles measured by degrees precisely, by simply advancing or receding for vertical angles, or moving to the right or left for horizontal ones, or a little varying the position of a station-staff; thus at once ensuring a simplified calculation and a more accurate result. The accuracy will also be augmented by some of the expedients which I shall explain as I go along.

The Table.

The table is so arranged, that for angles not exceeding 45 degrees, the

sine, cosine, tangent, cotangent, &c., for any number of degrees, will be found opposite the proposed number in the left hand column, and in the column under the appropriate word. When the number of degrees in the árc or angle exceeds 45°, that number must be found in the right hand column, and opposite to it in the columin indicated by the appropriate word at the bottom of the table. Thus, the siné and cosine of 36° are *58778 and 80902 respectively, the tangent and cotangent of 62° are 1-88073 and 53171 respectively; the radius of the table being unity, or 1.

The taking proportional parts for minutes, can only be done correctly (that is, independently of the rules of interpolation) in those parts of the table where the differences between the successive sines, tangents, &c., run pretty uniformly. In that case,

1. cosine

2. sine

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the mode to be employed will be evident from a single example. Suppose we want the natural sine of 20° 16. The sine of 21° is 35837, that of 20° 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 suni of 34202 and 436, or 34638 is very nearly the sine of 20° 16'. And so of others. But observe that the operation may often be contracted by recollecting that 10' are, 15' are, 40' are of a degree, and so on. Observe, also, that for cosines, cotangents, and cosecants, the results of the operations for proportional parts are to be deducted from the value of the required trigonometrical quantity in the preceding degree.

7. 1

cotan. = tangent.

In the annexed figure, where TH, BS, are horizontal, TB vertical, the latter is given=143: also angle HTS =35°, whence STB-90°-35° 55°. But BS is evidently the tangent of the angle STB to the radius TB. Hence BS TB x tan. STB=143 × 1-42815=204-22, agreeing with Dr. H.'s answer.

Ex. 3. Wanting to know the distance between two inaccessible objects NS (preceding figure) on a horizontal plane, the angles STB=641°, NTB=33°, were taken from the top of a tower whose height BT was known to be 120 feet. Required NS. (Hutton, p. 24.)

Here SN is evidently equal to the difference of the tangents of STB and NTB, to the radius TB.

Now, the differences of the tangents above 60° increasing rapidly, the common use of the proportional parts would give the tangent of 64° 30' too great. I therefore find the cotangent of 64° 30' 477 nearly, and from the principle of theor. 7, find 1477= 2.0964 tan. 64°. The tan. of 33° =64941. Therefore, (2.0964

64941) 120 = 144699 x 120 = 173.6388 SN. Dr. H.'s answer is 173 656, differing by about, the 10,000dth part.

Er. 4. Wanting to know the height of an inaccessible tower, I took two angles in the same vertical plane, yiz. N=58°, S=32°, (preceding

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figure) and measured the horizontal distance NS between the stations =300. Required the height BT of the tower, and my distance NB from it at the nearest station. (Hutton, p. 25.)

In this case, by simply reversing the process in the last example, NS divided by the difference of the tangents of STB and NTB, or the difference of the cotangents of S and N will give BT; and BT x cot. N=BN. But cot, 32°1-60033, and cot. 58°= 62487, their diff.=97546. Hence 30097546307·55 BT; and 307.55 × 62487=192∙17=BN, Dr. ̧ H.'s answers are 307-53 and 192.15; and the logarithmic method requires 12 lines, besides turning to several different pages of the tables.

Ex. 5. In order to find the height of a tower TB, that stood on the top of a hill, at two stations N, M, whose horizontal distance measured 200 feet, I took the vertical angles PNB =40°, PNT=51°, and PMT=33° 45', all in the same vertical plane. Required TB. (Hutton, p. 26.)

This example may be worked either by means of the tangents and cotangents, or by the sines; let us here employ the latter. Sin. MTN=sin. (TNP-TMP) sin. 17° 15'29654; sin. TMN sin. 33° 45'55556, found by the proportional parts, as before described; sin. B= sin. 50° sin. 130° 76604; sin. BNT=sin. (51° 40°) sin. 11°19081. With these numbers and MN=200, work the following proportions,

tances to be 735 and 840 yards, and the horizontal angle between those two lines 55° 40'. Required the distance. (Hutton, p. 27.)

The above will suffice to exemplify the manner of operation, as well as to prove the accuracy of the method; which, as will thus appear, is greater than I have hypothetically assigned it.

In what is here done, I shall not, I trust, be supposed attempting to supersede the use of the excellent tables referred to at the commencement of this paper, or the correct theoretical processes which they so greatly facilitate. I am solely anxious to explode all the crude and usually erroneous tentative methods adopted by those who are not conversant with the nature and use of logarithms, by showing that, without the aid of those artificial numbers, a table of natural sines, tangents, &c., comprehended in a single page, will enable a computer by simple operations in decimal arithmetic, to solve problems in trigonometry, and inaccessible heights and distances, with all the accuracy that can be desired by practical men.

The same table will be found of equal utility in the mechanical inquiries which relate to the parallelogram of forces, oblique pressures, motions on inclined planes, the usual practice of gunnery, &c. But fearing that I shall have already greatly encroached upon your pages, I can only hint at these applications now.

I am not, however, without hopes that what is here done will stimulate some individual of more leisure than myself to turn his attention to the abbreviation of a table of logarithms. I have seen such a table in a single sheet, and M. Wronski proposed to reduce it to a single page. If, however, a correct table to five places of decimals could be presented in about four pages, then two more pages might contain such a table as the preceding, and an analagous table of logarithmic