336 



THE POPULAR EDUCATOR. 



Fig. 55. 



THE DANISH BALANCE. 



This is a species of steelyard, in which (Fig. 55) the fulcrum 

 is movable, and the counterpoise is the weight of the beam 

 acting at its centre of gravity, p, the substance to be weighed 

 being suspended from a hook 

 or placed in a pan, at the ex- 

 tremity, E, on the other side of 

 the fulcrum. The question is, 

 how may yon graduate such an 

 instrument ? To do this, let us 

 suppose the beam to weigh 1 

 pound, and that 1 ounce of some 

 substance is placed in the scale ; 

 then it is evidentthatthefulcrum, 

 x , must be shifted to the point in which F R is to F F in the pro- 

 portion of 16 to 1, there being 16 ounces in the pound. This 

 comes to dividing the distance R p (which is known) into 

 seventeen equal parts, as proved in Lesson IV., and taking the 

 first point of division next to p for the fulcrum. If there be 2 

 ounces in the pan, RF must be to p p as 16 to 2 ; that is, you 

 divide B p into 18 parts, and take the fulcrum 2 from p. If 

 there be 7 ounces, you divide into 23 parts, and take 7 next to 

 P; and so on for all the ounces from 1 to 16 you may determine 

 the several positions of the fulcrum, marking them as you pro- 

 ceed. If the beam be of any other weight, you follow a similar 

 course, dividing B p into as many equal parts as there are ounces 

 in the sum of the weights of the beam and substance, and count- 

 ing off as many divisions from p as there are ounces in the 

 latter. 



From all this it is evident, first, that the subdivisions are not 

 equal to each other, as in the steelyard ; secondly, that the 

 operation of graduation is more troublesome than in that instru- 

 ment. The Danish balance, however, has the advantage of not 

 being encumbered with a movable counterpoise ; it carries its 

 own imperial standard weight within itself. 



THE BENT LEVER BALANCE. 



The principle of this instrument, a species of which is largely 

 sold for weighing letters, may be understood by the aid of the 

 accompanying Fig. 56. On an upright stand is placed a quadrant 

 arc, M o, of which c is the centre. Round c as a fulcrum revolves 

 a lever, usually bent, but in the figure represented as formed of 

 two arms at right angles to each other. The arm c B is gene- 

 rally of small weight, being lightly constructed, while the other, 

 CG, called the "index arm," is heavily weighted at its lower 

 end, the centre of gravity of the whole lever thus being nearly 

 ^c at some point, G, on that arm. 



On some substance, w, to bo 

 weighed, being suspended 

 from B, the index moves from 

 its zero point, o, up the quad- 

 rant until the weight of the 

 lever acting at G balances w 

 at B, that is, until the mo- 

 menta of these forces are 

 equal, which will be when w 

 multiplied by B H is equal to 

 1 the weight of the lever mul- 

 tiplied by G I. The divisions 

 of the quadrant corresponding 

 to the several weights 1, 2, 3, 4> etc., suspended from B are, 

 however, best determined by experiment for each weight. 



THE LEVER WHEN THE FORCES ARE NOT PARALLEL. 



In all the cases of levers and weighing instruments we have 

 eo far considered, the forces were supposed parallel in weighing 

 instruments necessarily so. The treatment of the subject is, 

 however, not complete until the condition of equilibrium is de- 

 termined for levers the forces acting on which are not parallel. 

 This is the most general case that can occur, and indeed it 

 includes all the others. To clearly understand it, let a lever be 

 defined a mass of matter of any shape which has one fixed point 

 in it. It may be a bar straight, or simply bent, or bent and 

 twisted, or it may be a solid block. So long as there is one 

 point fixed, we may treat it as a lever, that point being the 

 fulcrum. 



Moreover, the two forces which act on it are supposed to be 

 euch that their directions when produced meet, and that their 



Fig. 56. 



E 



Fig. 57. 



plane passes through the fulcrum. In cases where the two 

 forces do not meet, or their plane does not pass through the 

 fulcrum, there cannot be equilibrium. For example, the out- 

 stretched right arm of a man is a lever, of which the fulcrum is 

 in the right shoulder. Suppose, as he stretches it before him in 

 a horizontal position, one force is applied to the hand obliquely 

 from him towards the left to the ground, while another acts 

 horizontally at his elbow towards the right and at right angles 

 to the arm ; these forces cannot meet, and therefore would not 

 under any circumstances keep the arm in equilibrium ; further, 

 even were they to meet, they would not so keep it unless their 

 plane passed through the fulcrum in the shoulder socket. Sup- 

 I posing the forces, therefore, to be as described, namely, that 

 ; their directions meet and their plane passes through the fuicrun:. 

 j what is the condition of equilibrium ? In order that you may 

 olearly understand this, the knowledge of the following geo- 

 metrical principles is necessary. 



FURTHER PROPERTIES OF A PARALLELOGRAM AND TRIANGLB. 



1. Tlie area of a triangle is half that of any parallelogram which 

 has its base for one side, and a line drawn through its vertt . > 



to that base 'for tlie side opposite. This 

 appears from Fig. 57, where A v B is the 

 triangle, and A B c D any parallelogram 

 on A B formed by drawing from A and B 

 any two parallel lines A D, B c to meet the 

 parallel D c to A B through v. For, draw 

 v E parallel to A D, and therefore parallel 

 to B c, to meet A B in E. Then the triangle 

 A v B is made up of the two triangles A v E 

 and B v E. But since A E v D is a parallelo- 

 gram, the triangle A v E (Lesson III.) is 

 equal to A D v, and is therefore half the 

 parallelogram A E v D. So likewise is B v E 

 half B E v c ; and therefore the triangle A v B half A B c D. 



2. The area of a triangle is, in numbers, half tJie product of its 

 base and the perpendicular from its vertex on that base. This 

 follows from the previous principle. Let the number of inches 

 or feet, say inches, in A B (Fig. 58) be 6, and in the perpen- 

 dicular, v E, be 7, and construct on A B a parallelogram, 

 A B c D, whose sides are parallel to this perpendicular. Such a 

 parallelogram is termed a "rectangle," on account of its angles 

 being all right angles. Mark out the inches 



on A B and v E, and draw the dotted lines in 

 the figure parallel to A B and v E, cutting 

 this rectangle into the smaller ones the sides 

 of which are all equal to one inch, and which 

 are therefore so many square inches. Now 

 there are seven rows of these squares, one row 

 above the other, and there are six squares in 

 each row ; and therefore there are altogether 

 7 times 6, or 42, square inches in the rectangle. 

 But the triangle being half the rectangle, is 

 half of 42 square inches, that is, it is, in num- 

 bers, half the product of the base and perpen- 

 dicular. Were the numbers 13 and 9, or any other pair whatever, 

 the reasoning would be the same. 



3. If two triangles stand on opposite sides of a common base, 

 and the line joining their vertices is bisected by that base, the 

 triangles have equal areas. In Fig. 59, the triangles A B c, A B D 

 stand on the common base, A B, at opposite sides, and D c join- 

 ing their vertices is supposed to be bisected at M ; I have to 

 prove that the areas of the triangles are equal. Draw E F and 



H G through A and B parallel 



B G to D c, and also through D and 



c draw H E and G F parallel 

 to A B. Then we have a 

 large parallelogram E F G H, 

 which is divided into four 

 smaller ones by A B and D c. 

 But since D c is bisected 

 at si, making M c equal to 

 ill D, and therefore A E equal 

 to A F, the parallelograms 

 A F G B and A E H B are equal to each o'ther. But, as proved 

 above, the triangles ABC and A B D are half of these parallelo- 

 grams, and therefore are also equal to each other, as was 

 required to bo proved. 



Fig. 58. 



Fig. 59. 



