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If from a point C, of a triangle ABC, inscribed in a circle, there be a perpendicular CD let fall upon the opposite side AB ; that perpendicular is to one of the sides, including the angle, as the other side, including the angle, is to the diameter of the circle, i. e. DC: AC:: CB: CE.
Let the diameter CE be drawn and join EB; it is plain the angle CEB= CAB (by cor. 2. theo. 7. sect. 1.) and CBE is a right angle.(by cor. 5. theo. 7. sect. 1.) and=ADC : whence ECB=ACD. The triangles CEB, CAD, are therefore mutually equiangular, and (by theo. 16. sect. 1.) DC: AC :: CB : CE, or DC: AC:: CB : CE. Q. E. D.
Let three gentlemen's seats, A, B, C, be situate in a triangular form; there is given, AB 2. 5 miles, AC 2. 3, and BC 2. It is required to build a church at E, that shall be equi-distant from the seats A, B, C. What distance must it be from each seat; and by what angle may the place of it be found ?
Geometrically. By prob. 15. sect. 1. Find the centre of a circle
will be the place of the church; the measure of which, to any of these points, is the answer for the distance : draw a line from any of the three points to the centre, and the angle it makes with either of the sides that contain the angle it was drawn to; that angle laid off by the direction of an instrument, on the ground, and the distance before found, being ranged thereon, will give the place of the church required.
1. AB : AC+BC :: AC-BC: AD-DB,=516.
By cor. 2. theo. 14. sect. 1. The square root of the difference of the squares of the hypothenuse AC, and given leg AD, will give DC,=1.736.
Then, by the preceding lemma.
2. As CD: AC :: CB : the diameter,=2.65.
the half of which, riz. 1.325 is the semi-diameter, or distance of the church from each seat, that is, AE, CE, BE.
From the centre E, let fall a perpendicular upon any of the sides, as EF, and it will bisect it in E: (by theo. 8. sect. 1.)
In the right angled triangle AFE, you have AF, 1.15, and AE the radius 1.325 given, to find FAE,
3. As AF: R. :: AE : Sec. FAE,=29° 47'.
Wherefore directing an instrument to make an angle of 29° 47', with the line AC; and measuring 1.325 on that line of direction, will give the place of the church, or the centre of a circle that will pass through A, B, and C.
The above angle FAE, may be had without a secant, as before, thus ;
As AE: R. : : AF: S. AEF,=60° 18',
Its complement 29° 47', is FAE, as before.
The questions that may be proposed on this head being innumerable, we have chosen to give only a few of the most useful.
Containing a particular Description of the several
Instruments used in Surveying, with their res. pective Uses. And first,
OF THE CHAIN.
HE stationary distance, or mearings of ground, are measured either by Gunter's chain of four poles or perches, which consist of 100 links; (and this is the most natural division) or by one of 50 links, which contains two poles or perches; but because the length of a perch differs in many places, therefore the lengths of chains and their respective links will differ also.
The English statute perch is 54 yards, the twopole chain is 11 yards, and the four-pole one is 22 yards : hence the length of a link in a statute-chain is 7.92 inches.
There are other perches used in different parts of England, as the perch of woodland measures which is 6 yards; that of church land-measure, which is 7 yards (or the same with the plantation-perch) and the forest measure perch, which is 8 yards.
The Irish or plantation perch, is 7 yards, as before ; the two-pole chain is 14; and the four-pole one is 28
yards : hence the length of a link in a plantation chain is 10.08 inches.
The Scotch perch is 184 feet, or 67 yards, or 6 Scot's ells. In the shire of Cunningham in Scotland, their perch is 183 feet, and this perch is used in some few places in the north part of Ireland, as the statute perch is in some other parts.
For the more ready reckoning the links of a fourpole chain, there is a large ring, or sometimes a round piece of brass fixed at every 10 links; and at 50 links, or in the middle, there are two large rings. In such chains as have a brass piece at every 10 links there is the figure 1 on the first piece, 2 on the second, 3 on the third, &c. to 9. By leading therefore that end of the chain forward, which has the least number next it, he who carries the hinder end may easily determine any number of links : thus, if he has the brass piece number 8, next to him, and 6 links more in a distance, that distance is 86 links. After the same manner 10 may be counted for every large ring of a chain which has not brass pieces on it; and the number of links is thus readily determined.
The two-pole chain has a large ring at every 10 links, and in its middle, or at 25 links, there are two large rings ; so that any number of links may be the more readily counted off, as before.
The surveyor should be careful to have his chain measured before he proceeds on business, for the rings