given. And if any two angles of a triangle = 90°, then the remaining angle 90°, and the triangle is right-angled.

==

15. If one of the angles of an oblique-angled triangle be given, then that angle subtracted from 180° gives the sum of the other two angles; and if two angles are given, their sum subtracted from 180° gives the remaining angle.

16. If one of the equal angles of an isosceles triangle be found, then the other two angles are determined, because the sum of the two equal angles adjacent to the base subtracted from 180° gives the remaining angle at the vertex. And if the two angles at the base are equal, the sides that subtend them are equal, and the triangle is isosceles.

17. The two acute-angled triangles ABC and AER are each equilateral and equiangular, and therefore similar to each other, and each angle is 60°.

18. The three right-angled triangles ASB, ASC, and CGA are similar and equal to each other; ANE = ESA; also DBC

=

LAC MEC.

19. A triangle standing upon the chord of an arc with two sides radii is isosceles.

20. In the resolution of a right-angled triangle, as CAL, if c is taken for the centre of the circle, and CA for radius, then AL is tangent and CL secant. If L is taken for the centre of the circle, and LA for radius, then AC is tangent, and LC secant. And if L is taken for the centre, and LC for radius, then CA will be sine and AL = cosine.

21. To every arc measuring either an acute or obtuse angle two sines, two tangents, and two secants can, and often require to be drawn.

22. In an are subtending an acute angle the two sines and two tangents cross each other within the angle, so that the centre of the circle and the two points of intersection, the one where the two sines cross, and the other where the two tangents cross, are in a right line that bisects the arc, its chord, and the angle it measures.

23. Each of the two radii, which includes an acute angle at the centre, forms a portion of the two secants to an arc; and when the angle is 60° the portion of each that is inside the arc is equal to the remaining part outside the arc. Thus CB = BL, so that the tangent AL of 60° diverges from the arc AB, a distance equal to radius. 24. In an arc subtending an obtuse angle the two sines from its opposite extremities fall without the are perpendicularly upon the two secants, now portions of other radii than those which include the angle. The two tangents are limited by the two secants, ranging from the centre at an angle equal to the supplement of the arc, and

Y

in the opposite direction of the two radii that include the arc or angle. The secant therefore contains no part of the radii that include the angle, but the whole right line formed by each radius and its secant together is greater than the secant of its supplement by radius. Thus EL = CL+ EC.

25. In arcs subtending obtuse angles the parallelism of sines with tangents, and vice versa, lies on the opposite sides of the centre, the tangent being on one side, and its parallel sine on the other. Thus AS is parallel to EM, and EN to AL.

26. In arcs subtending acute angles the parallelism of the sines and tangents of an arc lies within the angle upon the same side of the centre. Thus AS is parallel to BD, and BV to AL.

27. The sines, tangents and secants of an arc are equal in magnitude or length to the sines, tangents, and secants of the supplement of that arc; but their position is different, so that a sine, tangent, or secant of the one cannot properly be termed the sine, tangent, or secant of the other, while the versed sine of an obtuse angle is greater than the versed sine of its supplement.

28. It is the direction and position of the two sides of a triangle that determine the measure of the angle they include, and vice versâ the measure of the angle they include that fixes the position of any two sides.

29. When a right line is drawn from the obtuse angle of an oblique-angled triangle, perpendicular to the base or side subtending the obtuse angle, it resolves the oblique-angled triangle into two rightangled triangles; the perpendicular thus drawn becomes the sine to each of the two angles adjacent to the base; the sides that include the obtuse angle are radii; and the segments of the base are each equal to the cosine of its respective adjacent angle. Thus in the obtuse triangle EAL, AS is sine to E and L, ES = cos. E, and LS = cos. L, and ASE and ASL are each right-angled triangles.

NOTE 1. The sine, tangent, and secant of an arc, are in some books defined to be the sine, tangent, and secant of the supplement of that arc. This, however, is not correct: the two arcs measure two angles of a different species, the one being acute and the other obtuse, according to their geometrical definitions (see Def. 12, 13, 15, 18, 19, 20, 22, 23, and 24, Part I.); consequently each has a separate and independent existence distinct from the other. Thus BD, although defined to be the tangent of AE is not its tangent; neither is EM the tangent of AB. Again BV is not the sine of AE, nor EN the sine of AB; much less is Es the versed sine of AE, viz. Es also the versed sine of AB; and lastly CD is not the secant of AE, nor CM the secant of AB. Hence the conclusion of dissimilarity as to position (cor. 27).

2. The diagram is drawn purposely to illustrate the dissimilarity of position that exists between the sines, tangents, secants, and versed sines of angles and their

supplements, and also to show some of the properties of triangles formed by lines in and about the circle that are of daily use in taking bearings and tie lines in surveying. It may also be added here that figs. 2 and 3 of the next section will still further show how absolutely necessary it is to attend to such distinctive characteristics as the position of lines, and the species of angles they subtend, in order to secure accuracy in the trigonometrical resolution of the triangles into which a survey may be divided, and to avoid ambiguity and error in plotting the same.

SECTION II.

ON SPECIES, CONSTRUCTION, RATIOS, RESOLUTION, AND SOLUTION, OF TRIANGLES.

1. Under Geometry, Part I., six kinds of plane triangles are defined, viz., (1) right-angled; (2) equilateral; (3) isosceles; (4) scalene; (5) acute-angled; and (6) obtuse-angled.

2. The angles of triangles are divided into three kinds, and defined (1) right angles; (2) acute angles; and (3) obtuse angles.

3. A plane triangle consists of six parts, viz., three sides and three angles; and if three of these, including one side, are given, the others, technically termed the sought parts, may be ascertained from the ratio or relation that subsists between them and the given parts.

4. In Plane Trigonometry triangles are divided into two classes, right-angled triangles and oblique-angled triangles. (The latter thus includes the five species of triangles that are not right-angled.)

5. In a trigonometrical survey the ground is divided into triangles. Thus any three station-poles not in a right line form a plane triangle.

6. Fields, farms, and estates may be divided and measured with the chain, either on the triangular method, as illustrated Plate III. Part V., or on the parallel method (Plate IV. Part V.), and yet both may be surveyed by the theodolite as if divided into triangles independently of those of the former survey. Thus the survey (Plate III.) is divided into two large triangles +1 +8 +10, and + 1 + 8 + 15; or into four triangles + 1 + 5 + 10, + 5 + 10 + 8, + 8 + 5 + 15, and + 15 + 5 + 1. Again the estate (Plate IV.) may be divided into two triangles + 1 + 6 + 32, and 1 +32 +40. Or the same station-poles shown on the plan may be used in taking the bearings, so as to give a much larger number of triangles in the trigonometrical survey. Thus each of the quadrilateral areas may be taken as forming two triangles.

7. In large surveys, where each of the interior triangles is bounded

by other three triangles, if the sides of one are determined, that will give one side in each of the adjacent triangles.

8. In trigonometrical surveying a side of one of the triangles is measured with the chain. This is termed the main-line or base-line. The other two sides may then be determined from the relation they bear to it (the measured line), and to the angles they subtend; and as a side of each of the adjacent triangles is thus found (by last article, 7), the sides of all the other triangles in the survey may therefore be ascertained from trigonometrical data.

9. The first triangle surveyed and plotted is the one that contains the base-line; and as this side is measured with the chain and the two adjacent angles with the theodolite, such data will give the other angle and species of triangle either by cor. 14, 15, or 16, Sect. I.

Right-angled triangles.

10. The field-book of the trigonometrical survey supplies only part of the dimensions requisite for plotting. Technically these are termed the "bearings," or "the given sides and angles," so that the remaining parts are found by calculation, or construction, or both, as illustrated in the following four examples.

Ex. 1. Let the base line be EB in the triangle AEB (fig. 1): let B = 60° and E 30°, then (by cor. 14) the triangle is right-angled and scalene. The side EB and angles B and E are given to find BA and EA: A is found by cor. 14.

Ex. 2. If the angles at the base are each 45° then EKB would represent the triangle; and as the angles E and B are equal, so are the sides BK and EK. The triangle is therefore both isosceles and right-angled, with a side and three angles given to find the remaining two sides.

Ex. 3. If one of the angles at the base is a right angle, then ASE would represent the triangle; ES the base line measured with the chain; E and s the angles taken with the theodolite; EA and SA the sides sought, and A the remaining angle, determined by cor. 14. Sect. I.

Ex. 4. In the interior triangles of a survey the two sides ES and SA, or EA and AS, or EA and ES, and the angle A = 90° may be found, when the remaining side EA and two acute angles E and A adjacent will be the parts sought or ES and the angles E and A; or AS and the angles E and A, as the case may be.

11. The large triangles on the plans Part V. Plates III. and IV. may be considered copies of those in the field, so that plotting the first two bearings is simply a work of copying (Problems XIX. XX. and

XXI. Part I.), and when the two bearings or angles at the base are given the first triangle may be formed (cor. 28). Thus draw the base line EB; next EA indefinite, making E = 30° (Problem XX. Part I.); BA in the same manner, making B = 60°: the two indefinite lines will, if the work is correctly performed, intersect each other at A, and form the triangle EAB right-angled at A. The lines EA and BA may then be measured by the scale on which EB has been laid down upon the plan, or they may be found by trigonometry. Thus, if EB is radius, as] in the first two examples, then BA would be sine and EA cosine. The ratio would therefore in these two examples be that of radius to the sine and cosine of the angle E = 30°. In the third example ES is radius, SA tangent, and EA secant, consequently the ratio is that of radius to the tangent and secant of E 30°. In the former case, either of the sides about the right angle may be sine or cosine; i.e., if the one is sine the other is cosine, and in the latter, either side about the right angle may be radius or tangent; ie., if the one is radius the other is tangent.

=

12. The trigonometrical solution of the four examples would therefore be effected by the following formulas :

Ex. 1. To find BA formula 1 Rad. : sin. E:: EB : BA

NOTE.-Under right-angled triangles it is common to consider the right angle as understood to be one of the parts given, although not so expressed. This however, is objectionable, for when articled pupils enter the field with such formulas stereotyped as it were upon their memories they are worse than useless to them, as they frequently lead to much confusion. In taking the bearings, right angles have to be found in the same way as acute and obtuse ones; and it may be further observed that unless the two acute angles = 90° as a proof check, rightangled triangles are treated as oblique ones; and as a general rule this is the more advisable one to follow in the trigonometrical survey. Thus the right-angled triangle EAB (Fig. 1) is resolved into two right-angled triangles by drawing the perpendicular As.