ON THE MEASUREMENT OF ANGLES. I. By degrees, minutes, and seconds. The right angle is divided into 90 equal parts, called degrees (°); a degree is divided into 60 equal parts, called minutes(); and a minute is divided into 60 equal parts, called seconds ("). Thus, 12° 15′ 36′′ denotes 12 degrees, 15 minutes, 36 seconds. II. By circular measure. Let BAC be an angle, and from the centre A describe the arc bc. If the radius of the circle be given, the angle BAC is measured by the length of the arc bc. B It is customary to call the radius of the circle unity; then the length of the semicircle, which is the measure of two right angles, or 180°, is 31416 = 3, approximately. When the angle is expressed in degrees, &c., the circular measure may be found by proportion. Ex 1. Find the circular measure of 54°. Ex. 2. 180: 54 :: 3 : 942. Find the circular measure of 23° 13′ 24′′. 180°648000", 23° 13′ 24′′ = 83604". 648000 83604 3: 405. 3. Find the circular measure of 52° 24′ 16′′. Ans. 915. 4. Find the circular measure of 29'42. Ans. '00856. Let ABCD be a rectangle of any convenient length and breadth, divided transversely into any number of equal parts by the lines ab, 11, 22, &c., and longitudinally into 10 equal parts by the equidistant lines 11, 22, 33, &c. Also let aB, bC be each divided into 10 equal parts in the points. 1, 2, 3, &c.; and draw a1, 12, 23, &c., and the scale is completed. By means of this scale we can measure the relative magnitudes of lines to two places of decimals. Thus, to take off the line represented by 3'7, set one foot of the compasses on 3, and extend to the number 7 in the division aB. To take off the line represented by 5.53, set one foot of the compasses on 5, and glide it down to the intersection of the line 33; then extend to the transverse line 67. This will give the required distance 5.63. When a number consists of 3 or more decimals, they may be all omitted after the second without material error. If the numbers are too large to be conveniently taken from the scale, they may be reduced by dividing them all by 10, 100, 1000, &c.; for since it is only the relative magnitudes of lines that we have to deal with, the ratio of 423 to 327 is the same as that of 42°3 to 327, or of 4°23 to 3'27; both of which can be taken from the scale. The latter number is represented by I. APPLICATION. To reduce a line of 40 inches to the standard of our scale, and divide it into 8 equal parts. Here the divisions are 5 inches apart; divide 40 and 5 by 10, and we have 4 and 5. Lay down the line AB = 4, and with the distance 5 in the compasses, step out the required divisions. 2°2, 2. To construct a triangle whose sides are 28, 22, 16. Divide each of these numbers by 10, and we have 2.8, 1.6. Lay down the base AB= 2.8, and from centres A, B, and radii 22, 16, describe arcs intersecting in C; join AC, BC. ABC is the triangle required. A 3. To construct an arch upon a given straight line. B Let AB be the given straight line; make AC equal to BD, and from centres C, D, at distance AC or BD, describe arcs; and from centres A, B, with the same radius, describe arcs intersecting the former in E, F; produce EC, FD to meet in G, and from centre G at distance GE, or GF, complete the arch AEFB. When the height of the arch and the span are given, the distance AC or BD may be calculated as follows: multiply the height of the arch by 1366, and the span by 183; the difference is the distance AC or BD. Example:-To construct an arch whose span is 36 feet, and height 12 feet. Divide 36 and 12 by 10, and we have 3.6 and 1°2; and the calculation gives AC= BD=98. The above figure is constructed from these numbers. The measuring chain is 22 yards long, and an acre is defined to be 10 square chains, or 22 x 22 x 10 = = 4840 square yards. Since the chain contains 100 links, the acre will contain 100 x 100 x 10 = 100,000 square links. A, R, P denote acres, roods, poles. SOLID MEASURE. 1728 Cubic Inches make 1 Cubic Foot. 27 Cubic Feet ......... I Cubic Yard. I Gallon contains 277 274 Cubic Inches. I Gallon of distilled Water weighs 10lbs. I Cubic Inch of Cast Iron weighs lb. nearly. EVALUATION. There are two kinds of numbers in arithmetic, namely, abstract, and concrete. An abstract number has no denomination, as 1, 2, 3, &c.; a concrete number has a denomination, as I pound, 2 feet, 3 shillings, &c. Abstract numbers can be multiplied together, as 3 times 4 is 12; a concrete number can be multiplied by an abstract number, as 3 times 4 shillings is 12 shillings; but we cannot multiply by a concrete number; for instance, we cannot say 3 shillings times 4, or 3 shillings times 4 shillings; but a concrete number can be divided by another concrete number of the same kind, as 12 feet contains 3 feet 4 times. Now, as the dimensions of geometrical figures are expressed in concrete numbers, a difficulty arises in limine in ascertaining their content or capacity; for arithmetic will not allow us to say 3 feet times 4 feet; but an intelligible meaning can be assigned from geometrical considerations. If the sides of a rectangle contain an exact number of yards, feet, inches, &c., and if through the points of division lines be drawn parallel to the sides of the rectangle, it is plain that the figure will be divided into a number of equal squares, each side of which will be 1 yard, I foot, I inch, &c.; and the number of such squares will be equal to the product of the numbers expressing the number of equal parts into which the sides of the |