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And, though in some cases, we have, with scant ceremony it may be, found our way along his passages, and picked his locks, yet this was in no slight degree for the sake of showing with what ingenuity both are constructed.
The cases in which a triangle is proved to be definite in shape and size (from its being proved equal to another triangle definite in all respects) are as follows :
I. When two sides and the angle between the two given sides are given, i.e. :
If AB, AC, and the angle CAB are A
Then CB and the angles ACB and CBA may be determined (1.)a
II. When the three sides are given,
If AB, AC, and CB are given;
Then CA, CB, and the angle ACB may be determined (3.)
In each of these cases the conditions are sufficient for the determination of the other parts. And how to determine the shape and size of the triangle, when any one set is fulfilled, is the subject of Trigonometry. This is of course beyond the scope of the present work, which is merely concerned with each case as far as it affects the position or arrest of a moving point.
a In these and the following figures, the given lines are in full, the dependent in dots.
CONSIDERATION OF THE CASES IN WHICH A MOVING POINT IS ARRESTED.
I. THE first case in which triangles are proved of definite shape and size is where the two sides and the angle between the two sides are given. That is, a point can lie but in one position when it is at a certain distance from a fixed point A, and in a certain direction in regard to it, (CAB.) (1.)
Again, when two points B and C are at definite distances from a third point A, and in definite directions with regard to it, then B and C are at a definite distance from each other. For example: York is about 200 miles north from London, and Bristol about 130 west of it; the distance therefore from Bristol to York must be ascertainable. But the most useful form of this proposition is, when the angle between the two given lines. is 90°. In this case, not merely has Geometry proved the distance to be ascertainable, but it has actually ascertained it. This is the problem of Pythagoras, which shall presently examine. And not only does it effect its own purpose, but being easily extended (Euclid II. 12 and 13,) to cases where the angles are of any given number of degrees, it serves at once as a basis for all measurement, as we shall attempt to show briefly before concluding.
II. The second case in which triangles are definite, is that wherein the lines forming it are given (2.;) that is, a point is fixed in position, if its distances from two fixed points be definite. Hence, if the
distances of any three points from each other be known, it is possible to determine their directions with regard to each other. This is a case of small practical importance and we pass on to the third.
III. This case is by far the most common in practice, and is only not the most important from the numerous results flowing from the problem mentioned as depending on the
first case. From it we learn, that a point is fixed in position when its directions with regard to two fixed points are definite (3.) Hence if it be possible to discover the direction in which any inaccessible point lies with regard to any two others of known distance from each other, it is possible to ascertain the distance of the first point also from either of the two latter. Now the directions of any point can always be ascertained by means of a theodolite, and some definite distance is always to be obtained, which is all that is requisite; so that the distance of any object can be measured by the means above described.
Thus, if we measure the angle which a line drawn to the centre of a planet (M) makes with the axis of the earth's orbit twice at six months' interval (4,) we have a triangle EME', of which the two angles MEE' and EE'M (which is 180°-ME'X) are known from measurement, and
the side EE'=190,000,000 miles,
that being the distance apart of the two positions of the earth at six months' interval. Hence by virtue of this proposition, E'M (that is the distance of the planet from the earth) is calculable also. When MEE' ME'X, the distance is infinite, as we saw in chapter xxii. and the star is called, not a planet (havýτns, moving), but a "fixed" star, as its distance is too great for the motion to be perceptible.
But the principle adapts itself with peculiar ease to measuring perpendicular heights, for here of course one angle (being 90°) is already known, and it is only necessary to measure a certain distance in any direction, and the angle at that point. Thus to know the height of a tower AB (5.)
Walk 100 yards from A to C, and measure with a theodolite the angle ACB (6,) then AB can be immediately ascertained.
One other curious example of this problem may be mentioned. All the above instances have required the use of a theodolite But suppose that a traveller, having no mathematical instruments with him, should arrive at the bank of a river, how is he to measure the distance across it? The place where he is
standing on the bank we will call A (9.) Then let him fix his
some object opposite to him, close to the water, a tree, a stone, or what not, and this we will call B. AB is the distance required. Next let him walk (say 50 yards) in a direction at right angles to AB, and there (C) put a small stick in the ground. Then having walked an equal distance further in the same straight line' he must turn off at right angles (of course away from the river) and walk on until the object at B is hidden by that at C, that is until he comes to a point (say E) which is in a straight line with B and C (10.) Then he will have two triangles, ABC and CED, in which
AC 50 yards=CD,
the angle BAC=90°=CDE,
and the angle BCA=opposite vertical angle DCE (for B, C, and E are in the same straight line).
.. By virtue of this proposition
i.e. the distance across the river at the point A is the same as the distance from D to E.
NOTE.-In fig. 6, A C, and in fig. 8, RC, ought perhaps rather to be in full than in dots; as though they are found indeed, it is by observation not by calculation, and they are therefore among the data, not the results of the problem.
WE now come to the following question. Since the distance of any two points from each other is definite, when the points are fixed in distance and direction with regard to a third, have we any means of ascertaining what the distance is. The answer has been