2. All the angles, which can be made at any point (by any number of lines), on the same side of a right line, are, when taken all together, equal to two right angles: and, as all the angles that can be made, on the other side of the line, are also equal to two right angles; therefore all the angles that can be made quite round a point, by any number of lines, are equal to four right angles. Hence also the whole circumference of a circle, being the sum of all the angles that can be made, about the centre, is the measure of four right angles.

3. When two lines intersect each other, the opposite angles are equal.

4. When one side of a triangle is produced, or extended, the outward angle is equal to the sum of the two inward opposite angles.

5. In any triangle, the sum of all the three angles is equal to two right angles (1800). Hence, if one angle of a triangle be a right angle, the sum of the other two angles will be equal to a right angle, (90°).

6. In any quadrilateral, the sum of all the four inward angles is equal to four right angles.

7. In any right-angled triangle, the square of the hypothenuse (or side opposite to the right angle) is equal to the sum of the squares of the other two sides. Therefore, to find the hypothenuse, add together the squares of the other two sides, and extract the square root of that sum: and to find one of the other sides, subtract from the square of the hypothenuse the square of the other given side, and extract the square root of the remainder for the side required.

8. cosine √(1

9. sine cosine = tangent. 10. cosine sine = cotangent. 11. sin.2+cos.2 = rad.2

12. rad.2 tan.2 secant.2

Thus, we may, instead of dividing by a sine, multiply by the cosecant; instead of dividing by a tangent, multiply by the cotangent of the same arc; and so of others.

6. perp. base = tan. angle at base.
7. base perp. = tan. angle at vertex.
8. hypoth.base = sec. angle at base.
9. hypoth. perp. sec. angle at vertex.
10. base hypoth. = cosine angle at base.
11. perp. hypoth. = sine angle at base.

TRIGONOMETRY, WITHOUT LOGARITHMS.*

"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, &c., 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."

* In Lieut.-Colonel B. Jackson's scientific "Treatise on Military Surveying &c., &c., &c.," Portable trigonometry without logarithms, is thus introduced"The following useful application of Trigonometry, by means of the natural sines, tangents, &c., is taken from an early number of that valuable periodical, The Mechanics' Magazine,' and will be found particularly suited to the purposes of the military surveyor."

A TABLE OF NATURAL SINES, COSINES, TANGENTS, COTANGENTS,
SECANTS, AND COSECANTS,

"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 tangent, and cotangent of 62 degrees are 1.88073, and 53171 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 200 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, 15 minutes are 1, 40 minutes are 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.

Example 1.-Having measured a distance of 200 feet in a direct horizontal line from the bottom of a steeple, the angle of elevation of its top, taken at that distance, was found to be 470 30', from hence it is required to find the height of the steeple?

By deducting 47° 30′ from 90°, the angle opposite the given side will be found (42° 30′).

Then by Case 1. TRIGONOMETRY :—

As sine
Or

42° 30′: 200 :: sine 47° 30':

·67556 : 200 :: •73723 : 208·2, &c., height required.

By construction

The triangle is constructed by making the side from a scale of equal parts, and laying down the angles from the protractor. Then by measuring the unknown parts by the same scale, the solution will be obtained.

Example 2.-Being on the side of a river, and requiring the distance to a house on the other side, 200 yards were measured in a straight line by the side of the river, and at each end of this base line the angles with the house were 680 2', and 73° 15'-required the distance from each end of the base line to the house?

The sum of the given angles (68° 2' + 73° 15′) subtracted from 180° will give the third angle (38° 43').

Then by Case 1. TRIGONOMETRY :—

As sine 38° 43': 200: sine 68° 2′

•62544: 200

92739: 296.5 first distance required.

As sine 38° 43′ : 200 :: sine 73° 15′

•62544: 200:: ·95753: 306·1 second distance required.

Similarly to the preceding examples, HEIGHTS, AND DISTANCES may be rapidly (and for military purposes, sufficiently accurately) computed in the field, by means of the foregoing trigonometrical table, if proper attention is paid to the principles by which the unknown angles of triangles may be ascertained: a base line, and requisite angle, or angles, having been given.

It will, however, be necessary to use advantageously the methods in Cases 1, (vide TRIGONOMETRY), and also the properties in the subsequent theorems, and corollaries."

TABLE,

showing the reduction in feet, and decimals upon 100 feet, for the following angles of elevation, and depression.

The reduction for 100 feet (from the above table) multiplied by the number of times 100 feet measured, will give the quantity to be subtracted from the measured length of an inclination, to reduce it to a horizontal position.

* For further information on Surveying, and Reconnoitring, reference should be made to the highly-valued publication, entitled "A TREATISE ON MILITARY SURVEYING, INCLUDING SKETCHING IN THE FIELD, PLAN DRAWING, LEVELLING, MILITARY RECONNOISSANCE, &c.," by Lieut.-Colonel Basil Jackson, containing a full account of every surveying instrument, and the right adaptation of them.