What length of board 8 in. broad is equal in area to a A. 17 in.

fquare ft. ?

If the time paft fince noon be equal to midnight; what time in the day is it?

of the time till

A. 20 min. past 5 o'clock.

A. So

Seven eights of what number is 6 more than of the fame ? mm 78 6 2 km Breast so fr HIERO, king of Syracufe, gave orders for a crown to be made of pure gold; but fufpecting the workmen had debafed it, he recommended the discovery of the fraud to the famous ARCHIMEDES; who in order to detect the impofition, procured a mass of pure gold, and another of filver or copper, each equal in weight to the crown; and each, being put feparately into a veffel fuil of water, expelled a certain quantity, which determined their specific bulks.

If the weight of each were 10 lb. and the water expelled by the alloy,92 lb. by the gold,52 lb. and by the compound crown,64 lb. what were the quantities of gold and alloy in the crown.?

Simples,925 2,-,64 mixture. A.,12 lb of alloy to,28 of gold or (x 25=) 3 parts alloy to 7 gold.

To what diftance would the lower end, of a bar of iron or wire 3 ft long, fink in a fountain of quickfilver; if the bar, being equal in all places, were kept perpendicularly and its specific gravity 7,645 and that of the quickfilver 13,6? As gravity of the quickfilver : that of the iron A. 1,626+ ft.

:: 3 :

If a chaise run 6 miles an hour, how much fafter does the top of the wheel move, than the bottom, or than that part which is 1 ft from the ground, the wheel being 5 ft high; and what length of road would it go while it turned a wheel, having 100 cogs, 20 times round, by moving it 5 cogs at each revolution (of the carriage wheel)?

As 2,5 ft heit. of axle: 6 velocity of axle :: 5 ft. top: 12 mi. an hour vel. of top (-0 vel. of bottom-12.) A. 5 times as fast as that 1 ft from bottom; and 5 ft x 3,1416 x 20 x 100-5 whole distance,

P

If A can do a piece of work in 10 days and B the fame in 13 days; in what time can both working together finish it? both can do 10+ 13 pieces in 10x 13 days A. 5 d.

What is the length of a pendulum, which will vibrate once in 3 feconds, and of one which will vibrate in fecond ?

3 x 3 x 39,2= A.,5x,5 x 39,2—A.=1 fec. pendulum. Five perfons depart from the fame place at the fame time; A travels 35 miles, B 30 eaft each day, C 50 fouth, D 30 weft, and E 30 north; how far are they from each other at the end of 7 days? A. A is 7 times 5 mi. from B, 35 x 7 and 30 times 7 from D, and “ from C and from 37874 3927 E, B is 30 times 7+ 30 times 7 from D, and the iquare root of that distance from C and E, &c.

If a bullet in 1 fecond fall from the top of a tree; what is its height? 147 qr x 7=49 ft. A,

"If 12 oxen eat up 34 acres of grafs in 4 weeks, and 21 oxen eat up 10 acres in 9 weeks; how many oxen will eat up 24 acres in 18 weeks, the grafs to be growing uniformly ?

3 1

36 oxen. A

The fquare root of the area of a circle is 21 of the cir cumference, or of the diameter, nearly; fo if the fquare 31 35 root of the area be divided by or,282 the qt, is the circumference, or by 31 or,886 the qt. is the diameter; or xy the circumference by for the square root of the area, &c,

31

What distance may the points of a pair of compaffes be ́alunder to form a circle containing a fquare foot or 144 in. and what distance must one point move to form fuch circle? A. 6,774+ extent of the points=di. and 42,58+ cir. What length of rope may be tied to a horfes head and the other end to a stake, to give him liberty of eating an acre of grafs ? 7,14+ rods. A.

GEOMETRY,

IS the fcience of quantity, extention or magnitude; and may be confidered as the art of reducing unknown to known quantities, and of comparing different lengths, fuperficies, and folids together.

*

GEOMETRICAL PROBLEMS.

To form a right angle or erect a perpendicular line.

Set one foot of the compaffes or dividers in a given point; as, e, form an arch ́c bd; then with one foot in d, bring the other to b, form an arch a c, and with one foot in a cross the arch at a, and a line from a to e is the perpen- d dicular, which forms the right angle at e.

Or with convenient radius, from any (giv en) point above the bafe, as a, for centre, form an arch croffing the bafe, bc; from centre c, form an arch at e or d, and from centre b, crofs the arch at e or d, and draw a line from a, as before for the perpendicular required.

Perpendiculars differently erected.

h

EXAMPLES,

a

Any thing whatever is its own perfect measure, yet its contents can not be comprehended or comparatively valued until it be reduced to some known quantity; as inches, feet, miles, acrs, gallons, pounds, &c. each of which has by use acquired a definite meaning; and though we may have a more accurate conception of the size of a thing by sight than by hearing of its contents in such kuown quantities; yet, its value, weight, &c cannot be so accurately computed, as by comparison with that whose value, weight, &c. is known.

To defcribe parallel lines. Form an arch of the fame fize at each end of a base, and draw a line from the top of one arch to that of another, for the parallel required.

To deferibe an ellipfe or oval. Form a small circle and fet one foot of the compaffes in one fide of the circle, with the other form she oppofit fide of the oval; fet one foot in i the part neareft to the ends of the oval to form the ends.

Or draw parallel lines across each other and defcribe arches as before from the points of meeting till they interfect.

Cr carry a ftring with the ends tied together round two fixed points; as a b.

To defcribe a circle which will pass through any three points, (as a b d) not in a fraight line. Set one foot of the dividers in each point and form an arch fo that the arches. croffed will point to the centre of the circle required; as, c. In the fame manner the centre of any given circle may be found.

semi.

20

To measure or form angles, by the divifions of a circle; as in T. 14. Extend the dividers 60° on any circle, or line of chords, as from O to ; fet one foot in the given 10. point, as b, with the other form an arch gf, and take the distance between the lines on the arch and apply it to the circle or line of chords, which will fhow the number of degrees the angle contains, as 40°. Subtract two angles of a triangle from 180° and the remainder is the third angle; as 4095 ·-180—45° e.

Any angle may be formed by taking the given number b

Line of equal parts. 1,625 in

40°

8,1

6 ၂ဝ

45°

90

of degrees; as 95° i in the dividers and fetting one foot in the arch or sweep of 60 as at g, then the other foot will reach to the place where the line ei should be drawn from the given point i to form the angle required, 95°.

Rright lines are measured by the equal divifions of a right line, and circular or curve lines by equal parts of a circle or by (the unequal parts of a right line) a line of chords, as from 0 to 180 in the preceeding femicircle.

TRIGONOMETRY,

By Geometrical Conftruction;

Is the art of protracting and measuring, with fuitable inftru- ments the fides, angles, &c. of plane triangles.

Three parts of a triangle must be given; viz. the three fides; two fides and one angle, or one fide and two angles to find the parts required. In a right triangle one angle is always given; viz. 90°

Two fides and an angle* between them given, to find the other fide, angles, perpendicular and contents.

RULE. Draw one of the given fides; as a b 6,5, then take the length of the other given fide in the dividers, as b c 3,75, and place one foot in the given point, b, with the other form an arch, cd, lay out the given angle; as b 80o 30', and draw the line to the arch c d from thence to the point a 7, is the fide required; the fhorteft diftance from the point to the bafe is the perpendicular ce, 3,7;..

measure the angles a c as before directed. The area may be found as under multiplication by multiplying the bafe and half the perpendicular together.

1 per.

2,3,7(1,85 x 6,5— 12,025 area.

d

6,5

80°30'

One fide and two angles given to find the other parts.

RULE. Draw the given fide its proper length, as a b 7; then lay out the angles from their proper points; as a 30%,

The angle given should be between the two given sides, otherwise there may be a mistake made, as frome to d; (for it is as far from b to one as to the other,) so we should be at a loss to know wheth en the side a c were 4,2 or 7, except the place of its touching the arch were determined, as at d or c..