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Fig. III. whereby you may find o P, (By Casė II. of R, A, P, T.)

OP (.

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thus :

As the Radius, Tang. 45 d. im

Is to the Side LP, 391.54. Foot ;
So is the Tangent of the Angle OL P, 22.25. d.
To the Height of the Hill OP, 160.18 Foot.
Which substracted from MP (the whole: Height) 335.93,

there remains 175.75 Foot, for the Alsitude of the Obje&t
M 0.

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Fig. IV.

IV. Horo the Altitude of the Sun may be taken, by the Shadow

of a Staff, or other Object, of a known Length.

Upon A B, being a level Plain, let there be erected a streight Staff, or the like, of any Length, fuppose 60.00 Inches, or 5 Foot) as CD; and the Sun Thining, suppose it casts the Shadotó thereof upon the plain Ground to E; which measured, suppose to contain 108 Inches, or 9 Foot.

Draw a Line at Pleasure, as A B; upon any Point thereof, as
D, erect a Perperdicular, upon which let 60 Inches, the length
of your Staff, from D to C; and the length of the Shadow there-
of 108 Inches, from D to É; drawing the Line of Umbrago CE,
and thus have you constituted a Right-angled Plain Triangle CDE,
in which there is given, (1.) The Leg. C D, 60 Inches. (2) The
Leg. D E 108 Inches, by which you may find the Angle Ć ED,
(by Cafe I. of R, A, S, T,) thus:
As the Length of the Shadow C D 108.00 Inches,

Is to the Radius, Tang. 45 d.
So is the Length of the Staff CD, 60.00 Inches,
To the Tangent of the CE D, 29.05 d.

And such is the Sun's Altitude at that time.
V. How the Height of an Accessible Object may be obtained,

by the Length of the Shadow of it.

Suppose that the Sun shɔuld cast the Shadow of some upright Obje£t, as F G, 84.5 Foot, from G to L. and at the same time, your Staff of 5 Foot, does cast its Shadow from L to K, 6.32 Foot: And from hence I would compute the Altitude of the Orji! FG.

Fig. V.


The Two Lines of the Obje& F G, and Staff H I, together with Fig. V. the Two Lines of Shadow, G L and LK, being laid down, do conftitute Two Right-angled Plain Triangles, viz. FGL, and HLR, Equiangled, and their Sides Parallel, and therefore Proportional (by Theorem VIII Lib. I.) And therefore, As the Length of the Shadow of the Staff LK, 632 Foot,

Is to the Height of the Staff H L, 5.00 Foot;
So is the Length of the Shadow of the Obje&G L 84.5 Foot,

To the altitude of the Object FG, 66.93 Foot.



. A to be

Of LONGIMETRI A. 1. How (standing upon an Object of a known Heighi) to find ibe Distance from thence, to some other remote Obječt.

the Side of a Fort or Bulwark 22.5 Foot high, Fig. VI. and being upon the Platform at C, you see a Tree, or other Objelt at B, whose Distance you would know, from the Foot of the Wall at A.

The Lines A B and A C being drawn, and the Height of the
Wall 22.5 Foot, fet from A to C, where by your Instrument di-
rected to B, you find the Degrees cut to be 71.25, which Angle lay
down, so have you the Right-angled Triangle C AB, in which
there is given, (1.) C A, the Height of the Wall 22.5 Foot.
(2.) The Angle observed at C, 71.25 Deg by which you may
tind the Distance A B, (by Case I. of R, Å, P, T.) thus:
As Radius, Tangent 45 Deg.

To CA, the Height of the Wall 22.5 Foot ;
So is the Tangent of A CB, the Angle observed, 71.25 Deg.
To the Distance AB, 66.28 Foot.
And if you would find the Length of the Visual Line CB,

you may (by Cafe V. of R, A, P, T,) thus: As the Sine of the Angle observed at C, 71.25 Deg.

Is to the Diftance B A, 66.28 Foot;
So is the Radius, Sine 90 Deg.
To the Visual Line C B; 69.98 Foot.


II. To take a Distance, ( Accessible or In-accessible) at Two Sias

tions. Fig. VII. 'Being in a field at E, there is a lVind mill (or other object, in

another Field at F, (separated by a River or other impediment) whose Distance is required.

In any other part of the Field, remote from E, cause a Mark to be set up, as at D; and measure the Distance between E and D, which liebe 115.00 Foot. — Then by your luftrument at E, find the Quantity of the Angle FE D, which suppose to be 106.50 Deg. And going from E to D, inake Observation of the Angle FDE, which suppose to be 57.10 Deg.

By these Two angles, and the Mensured Pistance, you have
constituted an Oblique Triangle D E F, in which there is given,
(1.) The Measured Dijtance E D, 11500 Foor. (2.) The Two
Angles at D and E, 106.50 Deg. and 57.10 Deg. And, (3.) The
Angle at F, (the Complements of the other T wo to 180 Deg.)
16.40 Deg. Whereby you may find the sides E F and D F, (by.
Axiom. II.) thus:
As the Sine of the Angle at F, 16.40 Deg.

Is to the Menfured Distance Е, 115.00 Foot;
So is the Sine of the Angle at F, 57.10 Deg.

To the Distance E F, 341.98 Foot.
And fo is the Angle at E, 105.50 Deg. (cr 73.50.)

To the Distance D F, 390.54 Deg.
III. If there ziere Three (or more) Ships on the Sea, and you be

ing upon the Land, desire to know how far those Ships are from

you; and also, how far they are distant one from the other. Fig. VIII. LESourcat M, are required to tell how far those Slips are from

ET the Three A; being upon you, and also, how far from each other.

First, Being at M, make choice of some other Place upon the Shoar, at some considerable Diftance, as at 0 130.00 Fathom.

Secondly, Being at M, cbserve the Quantity of the single AMO, which we will suppose to contain 104.50 Deg. and then removing to 0, and observing the Angle MO A, you find it to be 37.82 Deg: through which Degrees, Lines being drawn from M and 0, will cross each other in A, which is the Place of the first Ships which, with the Line of Distar.ce M 0, will form the


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Oblique-angled Triangle A MO: In which there is given, (1.) The
Side M O 130.00 Fathom. (2.) The Angle A M O, 104.50
Deg. And, (3.). The Angle A OM, 37.82 Deg. And having
Two Angles given, you have the third also given, viz. MA O,
37.68 Deg. whereby you may find the other Two Sides M A
and 0 A (by Axiom II.) thus:
As the Sine of MA O, 37.68 Deg.

Is to the Side M O, 130.00 Fathom:
So is the Sine of A OM, 37.82 Deg.
To the Side M A, 130,41 Fathom, the Distance of the Ship at

A, from M:
And so is the Sine of AMO, 104.50 (or 75.50 Dag.)
To the Side O A, 205.91 Fathom, the Distance of the Ship at
A, from 0.

Again, Observing from M and O, to B, you found the Angle B M O Fig. VIII. to contain 65.50 Deg. and MOB 68.00 Deg. wherefore, if upon M and O, you lay down those Angles, you shall have another Oblique-angled Triangle M BO: In which you will have given as in the former, (1.) The Angle BM 0, 65:50 Deg. (2.) BO M 68.00 Deg. And consequently M BO, 46.50 Deg, together with the measured Distance M ( 130 Fathom, by which you may find the other Two Sides M B and O B, by Axiom II. as in the former. For, As the Sine of MBO, 46.50 Degrees,

Is to MO, 130.00 Fathom:
So is to the Sine of B M0, 65.50 Degrees,

To the Side B (, 163.08 Fathom:
And so is the Sine of the Angle B OM 68.00 Degrees,
To the Side B M, 166.18 Fathom.

Observing again from M and O, to C, you find the Angle CMO
to be 11.75 Deg. and MOC 132.75 Deg. the which Angles laid
down, you have a third Oblique-angled Triangle CM0, wherein
there is given, as before, all the Three Angles, and One Side,
MO, whereby you may find the other Two, MC and OC
(by Axiom II.) as in the former..
So will Ń C be 164.39 Fathom, and OC 45.62 Fathom.



Fig. VIII.

And thus have you the Distances of all the Ships, from the Two places, M and o, on the Shoar.

Now for their Distances one from another:

And First, For the Distance A B.
In the Oblique Triangle A BM, there is given, the Sides A M
130 41 Fathom, and B M 168.18 Fathom, and the Angle contained
by them, A M B 39.00 Deg. whereby the third Side AB may be
found, (By Case II. of O, X, P, T,) thus :
As the Sum of the Sides, A M and B M, 296.59 Fathom,

Is to the Difference of those Sides, 35.77 Fathom:
So is the Tangent of half the Angles at A and B, 70.50 Deg.
To the Tangentof half the Difference of those Angles, 18.81 Deg.
Which added to 70.52 Deg: gives 89.31 D-g. for the greater

Angle B A M; and fubftracted therefrom, leaves 51.69
Deg. for the leffer Angle A B M.

Then say, (By Axiom II.)
As the Sine of the Angle A BM, :51.69 Degrees,
Is to the Side M A, 130.41 Fathom:

So is the Sine of the Angle A MB, 39.00 Degrees,

To the Side A B, 104.59. F.

And in the same manner may she Distance from B to C, and from C to A, be found.

According to this Method may the Distances of many Places upon the Land, one from another be obtained, by making of ObJervation, by a Theodolite, Semi Circle, (or other Graduated Infrument) from Two Places, from wherice all the other may be. seen : An Example whercof I shall give, with the manner of making the Observations; and protrašting of them, whereby the Triangles to be resolved for the Performance will be conspicuous: And for the resolving of them. (it being altogether the same with that foregoing). I fall leave to the Ingenuity of the Pra&ticioner.

Let A B C D E be feveral Places, as Churches in a Town or

City, or such like Objeds.
Fig: IX.

1. Make Choice of Two such Places, froin either of which, you may see all the Places whose Distances you require; which Places lét be F and G, distant from each other 1000 Foot, more or less.

2. Set

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