PART I. W Pl. I.

Fig. 8.

T

O lay down on paper, a rectilineal figure fimilar to a given rectilineal figure.

For example, fuppofe the given figure a quadrilateral, of which one fide is 235 feet, and the angle which it makes with the fecond fide 84°, and the fecond fide 288 feet, and the angle contained by it and the third fide 72o, and the third fide 294 feet. The part reprefenting a foot, is to be taken greater or less, according as you would have your figure greater or lefs. In this figure the icoth part of an inch is taken for a foot. Draw any ftraight line AB, and take 235 feet from the scale, and place it from A to B, which determines its length; at B make, by Prob. 3. an angle ABC of 84°, and make BC 288 parts, in the fame manner with AB. Then make the angle BCD 72o, and the fide CD 294 parts, and join AD, and it will complete the figure. And the angles at A and D can be measured as in the last problem, and the fide AD by the line of equal parts. In the fame manner may a figure be drawn fimilar to any given figure.

COR. Hence, any problem in Plane Trigonometry may be refolved by delineating the figure, as in this problem, and then measuring the unknown fides and angles.

Pl. I.

Fig. 9.

O defcribe the manner in which angles are meafured by the Quadrant.

Let the angle RAH, in a vertical plane, contained by the line AH parallel to the horizon, and AR coming from fome remarkable point of a tower or hill, or from the fun, moon, or a ilar, be the angle to be measured. Hold the quadrant with the limb DCB downward, fo that the eye at D may fee the object R through the fights in the fide AD, and the degrees and minutes in the arch BC between the line and plummet, and B the fartheft end of the limb will meafure the angle RAH. For from the make of the quadrant, DAB is a right angle, therefore BAR is a right angle, and CAH is a right angle, for AC is perpendicular to the horizon; therefore, taking away the common angie BAH, the remaining angle HAR is equal to the remaining angle BAC, of which the measure is BC. Hence the angle DAC is equal to the angle RAZ.

To take an angle of depreffion, as EAF. With the eye at A the centre of the quadrant, look through the fights on the fide AD

to

to the object F. Then the arch BC is the measure of the angle PART I. EAF; for each of the angles CAE, BAF being right angles, take away the angle DAC, which is common, and the angle BAC is equal to EAF.

PROB. VI.

To measure an acceffible height AB, by the square. Pl. I.

Fig. 10,

Let the inftrument lean on a fupport of a known height, and be directed towards A, fo that the fummet A may be feen through the fights, in one of the fides, as KL, and the plummet hanging freely, let it cut the fide KN neareft to the eye in N. Then, becaufe LN is parallel to AC, the angles KLN, KAC are equal, and LKN, ACK are right angles; therefore the triangles a 29. 1. E. LKN, ACK are equiangular; and NK is to KL, as KC to CA. b 4. 6. E. Suppofe the diftance CK 96 feet, and KN 80 parts, then, as 80 is

to 100, fo is 96 to 120; which therefore is CA.

b

But if the obferver be at G, fuch a distance, that the plummet line paffes through the angle P, oppofite to H the centre; then, because PG is equal to GH, GC is also equal to CA.

a

But let the distance BF or CQ be 300 feet, and let the plumline cut off 40 parts from the fide SI oppofite to the fights. Then the angle SIZ is equal to QZI, that is, to QAC and a 29. 1. .E. ZSI, QCA are right angles, therefore the triangles ZSI, QAC are equiangular ; and ZS is to SI, as QC to CA; that is, as c 3.1. E. 100 to 40, fo is 300 to 120; which therefore is the height of the object.

Note, If the height of a tower to be meafured end in a point, Fig. 11, as fig. 11. half the breadth or thickness BD of the tower is to be added to CD the distance of the observer from the tower,

to get his distance from the foot of the perpendicular AB,

PROB. VII.

To measure an acceffible height by the quadrant.

Find the angle C by the quadrant, as was directed in the 4th Prob. and measure CB. Then in the triangle ACB, right angled at B, are given CB, and the angle at C, to find BA: Wherefore, as R is to the tang. C, fo is CB to BA.

PROB

PART I.

Pl. I. Fig. 13.

T

O measure an acceffible height AB by means of a plane mirror.

Let the mirror be at C, in the horizontal plane BD; and let the obferver go back to D, till he fee the image of the fummit in the mirror, at a certain point of it, which he must carefully mark; and let DE be the height of the obferver's eye; then the angles DCE, ACB of incidence and reflection are equal, as is demonstrated in optics; and CDE, CBA are right angles; a 4. 6. E. wherefore, as CD to DE, fo a is CB to BA.

Pl. I. Fig. 14.

Note, Inftead of a mirror, the furface of water may be used.

PROB. IX.

O measure an acceffible height AB by means of

T two ftafis.

Let the longer staff DE be fixed perpendicularly in the ground, and move the fhorter one FG backward, until the fummit be feen over the ends F, D of the two staffs: and let DC, FH be parallel to the horizon, or to GB. Then the angle DFH is a 29. 1. E. equal to ADC; and DHF, ACD are right angles; therefore b 4. 6. E. FH is to HD, asb DC to CA; to which, if DE be added, the fum is AB.

Fig. 15.

a

Note, Many other methods might have been mentioned. For example, let EC be a ftaff erected perpendicular to the horizon, and EF its fhadow, and let BD be the fhadow of the height AB: Then, AD, CF being parallel, the angle CFE is equal to ADE; and CEF, ABD are right angles; therefore, as FE to EC, fo is DB to BA. And this is the case, if BD, EF be equally inclined to the horizon, as well as when they are parallel to it,

PROB. X.

To measure an inacceffible height AB by the Qua

drant.

Chufe the plane DC parallel to the horizon, and measure any diftance DC. in a ftraight line with the perpendicular AB; and at C take the angle ACB, and at D take the angle ADC: and 32. 1. E. because the angle ACB is equal to the angles CDA, DAC •,

the angle DAC is given; therefore, by the 3d Prop. of Pl. Trig. as the fin. DAČ to the fin. ADC, so is DC to CA, which

can

can therefore be found. And in the triangle ACB, right angled PART I. at B, by the 1ft Propofition of Plane Trigonometry, as R ton fin. ACB, fo is CA to AB.

T

PROB. XI.

O measure an inacceffible height AB, by means
of two staffs.

Pl. I. Fig. 17.

Let the obfervation be first made with the two ftaffs DE and FG, as in Prob. 8. then go off in a straight line from the height and first station, to a fecond station, and there place the longer ftaff perpendicularly at RN, and then the fhorter staff at KO, fo that the fummit A may be feen along their tops: Join KF, ND, and draw NP parallel to AF; therefore the angle KNP is equal to KAF; and AKF is common to the triangles NKP, a 29. 1. E. AKF; therefore, as KP: PN:: KF: FA : but because the tri- b 4. 6. E. angles PNL, FAS are fimilar, PN: NL :: FA: AS; therefore,

a

by equality, KP: NL:: KF : AS; therefore AS is found, to c 22. 5.E. which add SB the height of the fhorter ftaff, and the fum is the inacceffible height AB.

Note, In the fame manner may an inacceffible height be found by the square, or a speculum, or by the geometrical cross.

ROM the top of a given height AB, to measure Pl. I. the distance BC.

FR

a

Fig. 18.

Let the angle of depreffion DAC be taken, as in Prob. 4. and because AD is parallel to BC, the angle ACB is equal to a 29. 1. E. DAC; therefore, in the triangle ABC, right angled at B, are given AB, and the angle C, to find BC; wherefore, as the tangent C:R:: ::AB: BC.

Fig. 19.

To do the fame by the square, place it so that G may be feen Pl. II. through the fights, and obferve how many parts are cut off by the perpendicular: then the triangles AEF, ABC are fimilar; therefore, as EF: EA:: AB: BC 5.

b 4. 6. E.

O measure the diftance of two places A and B, Pl. II. one of which A is acceffible.

T%

Fig. 20.

PART I.

Fig. 21.

Take any ftation C, from which both A and B are feen, and measure AC. Place the theodolite at A, and direct the fights to B, and then to C, and mark the degrees and parts of a degree on the limb between them, and they will be the measure of the angle BAC: take, in the fame manner, the angle ACB; then the angle at B is known; therefore, by Prop. 3. Pl. Trig. fin. B: fin. C:: CA: AB, which will therefore be found.

To do the fame by the fquare, place the fquare horizontally at A, with the index on it, and direct the fights of the fquare along AB, and the fights of the index along the other fide of the fquare, and obferve through them fome acceffible mark C: then measure AC; and placing the fquare at C, direct its fights to A, and the fights of the index to B. If the index cut the fide RK adjacent to the fights, as at K, then CR: RK :: CA: a 4. 6. E. AB. But if the index cut the fide SL parallel to the fights, as at L, then LS: SC:: CA: AG, the distance fought, which will therefore be found.

Fig. 22.

Fig. 23.

Fig. 24.

To do the fame by four staffs. Fix a ftaff C, in the fame ftraight line with A and B, and draw the lines AH, CK parallel to one another, or perpendicular to CB; and place a staff at any point H of AH, and find the point K of CK, which is in a ftraight line with H and B, there fix a staff, and measure AH, CK; and if HG be parallel to CA, the triangles KGH, HAB are fimilar; therefore KG: GH or CA :: HA : AB.

Note, A ftraight line may be drawn in the field, perpendicular to another CH, from the point A, by making AB equal to AD, and fixing the ends of a line at B and D, the middle point of which has been previously marked; then let the line be ftretched by the middle E, until both parts of it are equally ftretched, and fix a ftaff at E: and the ftraight line AE is perpendicular to CH. In a manner fimilar to this, may any problem that can be refolved upon paper, by a scale and compaffes, be done by ropes or cords in the field.

A fhort distance may be measured thus: Draw AC perpendicular to AB; and from C draw CD perpendicular to CB, meeting BA in D: Then DA: AC :: AC: AB. Or AC may be a staff erected at A, and a mafon's fquare being put on C, direct one of the fides towards B, and mark the point D of the ground to which the other fide is directed.

PROB. XIV.

O measure the distance of two places, A and B, neither of which is acceffible.

The

Chufe