EXAMPLE V. Plate 5. fig. 3.

From the top of a tree 70 feet high, I took the angle of depreffion of two other trees, lying directly in a straight line from me, and on the fame horizontal plane with the tree on which I then stood, viz. that of the nearer 36°, and of the other, 55° 30'. Required their distance from the tree from which the obfervation was taken, and from one another.

Wanting to know the distance between a house and a tree, the tree being on the other fide of a river; I took my firft ftation at the house, and marked my fecond at B; the angle fubtended by the distance between my fecond station, and the tree is 60°. I then measured the distance between my first and fecond stations, 380 yards, and found the angle fubtended by the house and tree to be 43°. Required the distance between the house and the tree.

As

EXAMPLE VII. Plate 5. fig. 5.

I wished to know the diftance between a kirk and a mill, which were upon the other fide of a river, I choose two ftations, A and B, diftant 400 links, and found the angles MAK 40°, KAB 64° 25', and ABM 56° 15', MBK 50° 8'. Required the distance between K the Kirk, and M the Mill.

In the triangle AKM, to find the angles AMK, MKÁ.

21 37

13.58350 the lefs

To tan. AMK-MKA 48° 23'10. 05138

Note, The foregoing example may be performed, by ufing MB and BK as the containing fides.

EXAMPLE VIII. Plate 5. fig. 6.

If the Peak of Teneriff be four miles above the level of the fea, and the angle of depreflion taken from the fartheft vifible point, be 87° 25' 55". Required the diameter of the earth, alfo the fartheft vifible point that can be feen from the Peak.

If the fquare of the visual ray, being a tangent to the earth, be divided by the height of the spectator's eye, above the level of the fea, the quotient will give the earth's diameter, and the height of the spectator's eye above the level more.

Demon. Because the straight line AC is equally divided at E, and produced to the point D, the rectangle AD, DC, together with the fquare of EC, is equal to the square of ED, but the fquare of ED is equal to the squares EB, BD, because DBE is a right angle; therefore, the rectangle AD, DC, together

with the fquare of EC-EB, is equal to the fquares EB, BD; take away the common fquare EB, and the remaining rectangle AD, DC, is equal to the fquare of BD the visual ray. And because the rectangle AD, DC, is equal to the square of BD, (Euclid. 17th. 6.) DC: DB::DB: AD.: Therefore, DB2 AD and AD-DC-CA the diameter.

Here it must be obferved, that if from any point without a circle, two ftraight lines be drawn to touch the circle, they are equal to one another, (Eucl. 37. 3.); therefore, FC is equal to FB, but BF and FD make up BD the visual ray; confequently, it will be 89.18+89.27=178.45=BD, and 178.452=7961

4

=AD, and 7961—4=7957, the earth's diameter nearly.

Several methods have been invented to find the earth's diameter. Mr Picart of the Academy of sciences at Paris, has proposed an exact method, by which, not only the equatorial and polar diameters may be known, but also the figure of the earth determined.

According to Mr Picart, a degree of the meridian at the latitude of 49° 21', was 57.06 French toifes, each of which con tains 6 feet of the fame measure; from which it follows, that if the earth be an exact sphere, the circumference of a great 'circle of it, will be 123.249,600 Paris fect, and the semidiameter of the earth, 19.615,800 feet: but the French ma'thematicians, who, of late, examined Mr Picarts obfervations, • affure us, that a degree in that latitude, is 57.183 toifes. They measured a degree in Lapland, in the latitude of 66° 20′, and found it to be 57.438 toifes. By comparing thefe degrees, as well as by the obfervations on pendulums, and the theory of gravity, it appears, that the earth is an oblate fpheriod; and the axis or diameter that paffes through the poles, will be to the diameter of the equator, as 177 is to 178, or the earth 'will be 22 miles higher at the equator, than at the poles. A degree has likewife been measured at the equator, and found 'to be confiderably lefs than in the latitude of Paris, which confirms the oblate figure of the earth. Hence it appears, that if the earth were of an uniform denfity from the surface to the centre, then according to the theory of gravity, the meridian would be elliptical, and the equatorial would exceed the polar diameter, by about 44 miles.'

PROBLEM III. Plate 5. fig. 9.

To find the height of an object, by means of one staff.

Suppose the pole AB of an unkown height, BC a horizontal plane, and ED a staff of a known length. At any conve