ent height, by drawing parallel lines; divide the altitude of these rectangles into ten equal parts, and, through each of these parts, draw parallel lines the whole length of the scale. Divide the firft divifion AB into ten equal parts, also CD into as many, and connect these points of divifion by diagonal lines, and the scale is finished.

In taking measures from the diagonal fcale---If the large divifions be reckoned units, the small divifions from A to B will be decimals. If the great divifions be 10, each of the small divifions is an unit; and if the great divifions be 100, then each of the fmall divifions is 10, and each divifion in the altitude is an unit.

If it were required to take off 456 from the fcale; with one foot of the compaffes on 4, extend the compaffes till you have 4 of the great divifions and 5 of the leffer; then flide up your compafles with a parallel motion till you come to 6 on the pa rallel lines, and you have the extent required.

PROBLEM XXVIII.

The conftruction of the line of chords, fines, tangents, and fecants.

About the centre C, with any convenient radius *, describe the femicircle ADB; erect the perpendicular CF, which will divide the femicircle into two quadrants, viz. AD, BD: divide the quadrant DB into nine equal parts, and upon the point B erect a perpendicular BT, then draw AD and BD.

On B as centre, transfer each of these divifions in the quadrant DB, to the straight line BD; then is BD a line of chords.

From the points 10, 20, 30, &c. in the quadrant BD, drop. perpendiculars upon the diameter AB; transfer the perpendi-. culars

The degrees are numbered from B to D.

eulars to DC; so shall DC be a line of fines, and CB a line of verfed fines.

From the centre C, draw ftraight lines through each division in the quadrant BD, to meet the tangent BT; fo fhall BT be a line of tangents.

From the centre C, with the distances of each of the lines which meet the tangent, fweep arches to cut CF; then fhall CF be a line of fecants.

If from the point A ftraight lines be drawn to the several divifions in the quadrant DB, they will divide the radius CD into a line of femitangents.

Again--Divide the quadrant AD into eight equal parts, and from A, transfer the divifions to the line AD; then fhall AD become a line of rhumbs, each divifion anfwering to a point of the mariner's compafs.

PROBLEM XXIX.

The angles, and one leg of a right-angled triangle being given, to conftruct the figure, and find the other leg.

Angle A=30° 40′

Given Angle C=59° 20' Required BC.

AB=300

From the diagonal fcale make AB 300; upon B erect a per pendicular of an indefinite length; and at the point A make an angle of 30° 40′; then draw the line AC, and it is done. If the angle at C be measured, it will be 59° 20'; and if the leg B C be applied to the fame diagonal fcale from which AB was taken, it will measure 177.9. Plate 3. fig. 43.

PRO

PROBLEM XXX.

The hypothenufe and all the angles being given, to find the legs,

AC=568

Given Angle A=39° 14' Required AB, and BC.
Angle C-50° 46′

Draw the line AB of an indefinite length, and draw AC equal 568, making with AB an angle of 39° 14′; and from C drop a perpendicular, cutting the bafe in B, and it is done: For if angle C be measured from the line of chords, it will meafure 50° 46'; and if AB be measured from the fame diagonal fcale, it will measure 440, also. BC 359.2. Plate 3. fig. 44.

PROBLEM XXXI.

The two legs of a right-angled triangle beeng given, to find the acute angles, and the hypothenufe.

fAB=150 Req. angle A, angle C,.
BC=160 and AC.

Given {

=

From any diagonal fcale, draw AB=150, and from the fame. fcale draw BC perpendicular to the former 160; join AC, and the triangle is constructed: for if angle A be measured from a line of chords, it will be 46° 51'; alfo angle C 43° 9'; and AC will be 219.3 equal parts. Plate 3. fig. 45.

PROBLEM XXXII.

The hypothenufe and one of the legs being given, to find the acute angles and the other leg.

Given {BC=150} Required ang. C, ang. A, and AB.

90.

Draw

Draw the base AB, upon B erect the perpendicular BC e qual 69; take 150 from the fame scale, and with the centre C, and radius 150, defcribe an arch to cut the bafe in A; join AC, and it is done: For angle A will measure 27° 23', and angle C 62° 37', and the bafe BC 133 equal parts. Plate 3. fig. 46.

PROBLEM XXXIII.

Given two angles of an oblique angled triangle, and the fide oppofite to one of them; to find the other fides.

Find the fupplement of the fum of the two given angles, thus: 59° 0'+ 52° 15=111° 15'.

And from 180° fubtract 111° 15', the remainder will be, 68° 45'; then draw AB equal 276.5: Draw AC, making angle A 59°, and from B draw BC, making angle B 68° 45', and meeting AC in the point C, and it is done: then shall AC measure 325.9, and BC 299.7. Plate 3. fig. 47.

PROBLEM XXXIV.

Two fides of an oblique angled triangle, and the angle oppofite to one of them being given, to find the other angles and the third fide. _

Given

AB-26
AC=39.42

Ang. B=91° 15')

Required ang. A ang. C and BC.

Draw the bafe AB equal 26, and at the point B make an angle of 91° 15′ by BC; then on A as centre, with the ra

dius 39.42, describe an arch cutting BC in C, and join AC,

and it is done.

So fhall angle A measure 47° 30', and angle C 41° 15′; alfo BC 29.07 equal parts. Plate 3. fig. 48.

PROBLEM XXXV.

Two fides, and the contained angle of any triangle being given, to find the remaining angles, and the third fide.

Given

AC=60
BC=50
Ang. C=45°.

Required ang. A ang. B and AB.

Draw AC equal 60, and BC equal 50 equal parts, meeting in C at an angle of 45°; then join AB, and it is done: For if you take AB in your compaffes, it will measure 43.1 on the fame fcale of equal parts; alfo angle A will measure 55° 7′, and angle B 79° 53', from the line of chords. Plate 3. fig. 49

LOGA