WEDGE
, in Geometry, is a solid having a rectangular base, and two of its oppo- site sides ending in an acies or edge. Thus, AB is the rectangular base; and DC the edge; a perpendicular CE, from the edge to the base, is the height of the Wedge. When the length of the edge DC is equal to the length of the base BF, which is the most common form of it, the Wedge is equal to half a rectangular prism of the same base AB and height EC; or it is then a whole triangular prism, having the triangle BCG for its base, and AG or DC for its height. If the edge be more or less than AG, its solid content will be more or less. But, in all cases of the Wedge, the following is a general rule for finding the content of it, viz,
To twice the length of the base add the length of the edge, multiply the sum by the breadth of the base, and the product by the height of the Wedge; then 1/6 of the last product will be the solid content.
That is, the content. See this rule demonstrated, and illustrated with examples, in my Mensuration, p. 191, 2d edition.
Wedge, in Mechanics, one of the five mechanical powers, or simple engines; being a geometrical Wedge, or very acute triangular prism, applied to the splitting of wood, or rocks, or raising great weights.
The Wedge is made of iron, or some other hard matter, and applied to the raising of vast weights, or separating large or very firm blocks of wood or stone, by introducing the thin edge of the Wedge, and driving it in by blows struck upon the back by hammers or mallets.
The Wedge is the most powerful of all the simple machines, having an almost unlimited and double advantage over all the other simple mechanical powers; both as it may be made vastly thin, in proportion to its height; in which consists its own natural power; and as it is urged by the force of percussion, or of smart blows, which is a force incomparably greater than any mere dead weight or pressure, such as is employed upon other machines. And accordingly we find it produces effects vastly superior to those of any other power whatever; such as the splitting and raising the largest and hardest rocks; or even the raising and lifting the largest ship, by driving a Wedge below it; which a man can do by the blow of a mallet: and thus the small blow of a hammer, on the back of a Wedge, appears to be incomparably greater than any mere pressure, and will overcome it.
To the Wedge may be referred all edge-tools, and tools that have a sharp point, in order to cut, cleave, slit, split, chop, pierce, bore, or the like; as knives, hatchets, swords, bodkins, &c.
In the Wedge, the friction against the sides is very great, at least equal to the force to be overcome; because the Wedge retains any position to which it is driven; and therefore the resistance is at least doubled by the friction.
Authors have been of various opinions concerning the principle from whence the Wedge derives its power. Aristotle considers it as two levers of the first kind, inclined towards each other, and acting opposite ways. Guido Ubaldi, Mersenne, &c, will have them to be levers of the second kind. But De Lanis shews, that the Wedge cannot be reduced to any lever at all. Others refer the Wedge to the inclined plane. And others again, with De Stair, will hardly allow the Wedge to have any force at all in itself; ascribing much the greatest part to the mallet which drives it.
The doctrine of the force of the Wedge, according to some writers, is contained in this proposition: “If a power directly applied to the head of a Wedge, be to the resistance to be overcome, as the breadth of the back GB, is to the height EC; then the power will be equal to the resistance; and if increased, it will overcome it.”
But Desaguliers has proved that, when the resistance acts perpendicularly against the sides of the Wedge, the power is to the whole resistance, as the thickness of the back is to the length of both the sides taken together. And the same proportion is adopted by Wallis (Op. Math. vol. 1, p. 1016), Keill (Intr. ad Ver. Phys.), Gravesande (Elem. Math. Lib. 1, cap. 14), and by almost all the modern mathematicians. Gravesande indeed distinguishes the mode in which the Wedge acts, into two cases, one in which the parts of a block of wood, &c, are separated farther than the edge has penetrated to, and the other in which they have not separated farther: In his Scholium de Ligno findendo (ubi supra), he observes, that when the parts of the wood are separated before the Wedge, the equilibrium will be when the force by which it is pushed in, is to the resistance of the wood, as the line DE drawn from the middle of the base to the side of the Wedge but perpendicular to the feparated side of the wood continued FG, is to the height of the Wedge DC; but when the parts of the wood are separated no farther than the Wedge is driven in, the equilibrium will be, when the power is to the resistance, as the half base AD, is to its side AC.
Mr. Ferguson, in estimating the proportion of equilibrium in the two cases last mentioned by Gravesande, agrees with this author, and other modern philosophers, in the latter case; but in the former he contends, that when the wood cleaves to any distance before the Wedge, as it generally does, then the power impelling the Wedge, will be to the resistance of the wood, as half its thickness, is to the length of either side of the cleft, estimated from the top or acting part of the Wedge: for, supposing the Wedge to be lengthened down to the bottom of the cleft, the power will be to the resistance, as half the thickness of the Wedge is to the length of either of its sides. See Ferguson's Lect. p. 40, &c, 4to. See also Desagu. Exp. Phil. vol. 1, p. 107; and Ludlam's Essay on the Power of the Wedge, printed in 1770; &c.
The generally acknowledged property of the Wedge, and the simplest way of demonstrating it, seem to be the following: When a Wedge is kept in equilibrio, the power acting against the back, is to the force acting | Perpendicularly against either side, as the breadth of the back AB, is to the length of the side AC or BC. —Demonstra. For any three forces which sustain one another in equilibrio, are as the corresponding sides of a triangle that are drawn perpendicular to the directions in which the forces act. But AB is perpendicular to the force acting on the back, to drive the Wedge forward; and the sides AC and BC are perpendicular to the forces acting upon them; therefore the three forces are as the said lines AB, AC, BC.
Hence, the thinner a Wedge is, the greater is its effect, in splitting any body, or in overcoming any resistance against the side of the Wedge.