The ancient balance was the statera, or steelyard, in which the arms are of unequal length, and one movable weight is used, placed at different distances from the centre of motion, or fulcrum. The annexed figure represents the common steelyard, in which c is the fulcrum, or centre; c b the longer arm, and c a the shorter, e is the article to be weighed, suspended to the shorter arm, and d the constant weight. Now, if the shorter arm, by its additional thickness, be a counterpoise to the longer, so that the beam, when unloaded, may hang in a horizontal position, it is manifest, that equal weights, hung at equal distances from the centre, will balance each other; but if one of the weights be removed further from the centre, that side will preponderate. From this, it appears that a large weight e suspended at b, may be counterpoised by a small one suspended at d. If the distance between c and d be nine times greater than that between c and e, a weight of 10 lbs. at d will counterbalance one of 90 lbs. at e.

To prevent the necessity for calculation, the longer arm is graduated, so that the exact weight may be known by inspection.

The Danish Balance is also a steelyard, but in this the weight is fixed at one end, the article to be weighed at the other, and the fulcrum, or support, movable between them. In the annexed cut, b represents the standard weight, and a the hook, to which the article whose weight is required, may be suspended, and d is the movable fulcrum. If a b were supposed a perfectly straight rod without weight, the gradations on it should be at equal distances; but as this cannot be the case in practice, a different arrangement is required. If c be the centre of gravity of the beam, and the fulcrum be placed at this point, it is clear that the beam will be supported in an horizontal position; but if a weight be appended to the hook a, the centre of gravity, which in all cases must be supported, will be removed to d, for example, and, consequently, the fulcrum must be moved to the same point. In this case, there is not only a difference in the leverage, or length of the arms, but there is the weight of the portion c d taken from one side and added to the other.

The best method of graduating this instrument is by experiment, - by applying known quantities at the point a, and marking the place of the fulcrum d when an equilibrium takes place.

The Chinese Balance is a steelyard, somewhat different from the Roman statera. It is much used by the Eastern merchants in weighing gems and precious metals. The beam is a small rod of wood or ivory, about a foot in length. Upon this there are three lines of measure, made of delicate silver studded work. The scales commence at the end of the beam, whence the first extends to 8 inches; the second to 61/2; and the third to 81/2. The first indicates European weight, and the other two Chinese. At the other end of the beam a scale is suspended; and at three several distances from this end are fastened so many fine strings, forming so many different points of suspension. The distance of the first point from the end is 8/5 of an inch; the second 16/5; and the third 24/5. When the instrument is used, it is hung up by one of the strings, and a sealed weight of about 11/4 oz. is hung upon some one of the divisions of the rule, so as to counterbalance the weight of the article, which is indicated by the graduations of the scale.

The Bent Lever Balance is represented in the annexed figure, in which a c b is a bent lever, moving on the centre c as its fulcrum, or axis, of motion. To the shorter arm of the lever at b, a scale-pan e is appended, while the other arm has a heavy weight affixed to its other extremity a, which passes over the quadrantal arch f g. The substance to be weighed being placed in the scale e, the end a will indicate the weight by the height to which it rises on the graduated arch. A little attention to the diagram will show that, as the end b descends, the other extremity a ascends, and, at the same time, removes to a greater distance, from a vertical line passing through the centre of motion. In the present position of the balance, the effective length of the arm c b, is k i, and of the arm c a, is k d. Now, as these are of equal lengths, the weight a (omitting the weight of the lever itself) will be equal to that of the substance placed in the scale. But as the weight approaches the pointy, the effective length of a c will be represented by k h, and the weight a will therefore act with as much more power, as the length of k h exceeds that of k d.

If the point continued at the same distance from the vertical line, passing through c, the efficacy of the weight would, at any point in the arch, be proportional to the length of a perpendicular drawn from that point to the vertical line; but as the distance of b is constantly varying, we can only state, generally, that the divisions will be nearer together as we approach the upper part of the scale.

Payne's Weighing Machine is of the steelyard kind; the longer arm is divided by lines denoting the various weights, as usual; but, instead of the weights being suspended by hooks immediately on the beam, they are attached to a long case or box, which slides with some friction along the beam. Beneath one end of this sliding box is a large hook, to which is suspended the heavy weight, which is used to measure the larger quantities, as hundred weights, and quarters, which are denoted by the divided lines on the beam, as the sliding box is drawn over it. To measure the smaller quantities, as pounds and ounces, there is a light scale of parts fixed to the top of the sliding box, to which a hook and weight are hung, which are applied in the same manner as the common steel-yard. In the preceding engraving we have given a view of the whole arrangement, the longer arm of the beam being somewhat shortened to save room. In this representation, a a is the beam; c is the fulcrum; d a long rectangular loop through which the arm a a passes, and which serves to support it when not in use, or to limit its vibrations when employed in weighing; e is the sliding-box, with its graduated scale, for the minuter quantities, which are to be ascertained by the smaller weight g; f the larger weight, which may be secured at pleasure, at any point, by means of a thumb-screw above, half a turn of which fixes the slide against the beam, while the more minute quantities are being taken.