RECTANGLE (or right angled), the name given to any figure of which all the angles are right angles. Hence the figure having as many right angles as sides in the sum of its angles, must be foursided; for none but a foursided figure has the sum of its angles equal to four right angles. It is unnecessary to give ,a diagram of the most common of all the forms of art ; the page of this book may serve as an instance.
The properties of the rectangle, to which it owes its importance in a mathematical point of view, consist of one which it shares in common with all parallelograms, and one which marks it as the most simple of parallelograms. Every parallelogram, and the rectangle anion., the rest, may be divided in an infinite number of ways into parallelo grams having the same angles as the original parallelogram ; and if any parallelogram be divided into others by lines drawn parallel to one only of the sides, the smaller parallelograms bear to the whole the same ratios which their several bases bear to the whole base. Also the area of a rec tangle may be immediately deduced from nothing but the length of its two sides. If as a superficial unit we choose a rectangle having the sides A and B, it may immediately be told how many times and parts of times any other rectangle contains the unit: Measure one side, and see how many times it contains A (say 2!); measure the other side, and see how many times it contains B (say 3!) ; then the product of 21 and 3?, or 8 23 184 16 or , or 8 H, is the number of times which the rectangle to be measured contains the unit rectangle. This may be shown as follows :— Let r Q a s be the rectangle to be measured, and try the unit rectangle, P IT being A, and r 'r being B. The rectangle me Q is so drawn that P a contains 21 of A, and P Q contains 3# of B. The whole rectangle is obviously divided into six rectangles of the size of r v : six at the top, each of which is one-third of r v ; four on the right, each of which is one-seventh of r v ; and four higher up on the right, each of which is the twenty-first part of r v. We have then, on the whole, r v repeated In practice it is most convenient to make r x and r u equal to one another, and equal to the unit used in measuring lengths. Hence the rule for finding the area of a rectangle is : multiply together the number of linear units in the two sides, and the result is the number of square units (or squares on the linear unit) in the rectangle. This rule is abbreviated as follows : the product of the sides of a rectangle is the area ; an abbreviation which often confuses the mind of a beginner, who imagines that two lines can be multiplied together [MuurirmeAssort], and that the rectangle, that is, the very shape of the rectangle, is the product; a mistake precisely that of a person who should imagine that the very silver of ten shillings could be multiplied by seven yards of stuff, and that the product could be seventy shillings. Now seven yards of stuff at ten shillings a yard certainly cost as many shillings as there are unite in 7 x 10 ; and a rectangle whose sides are seven and ten feet certainly contains as many square feet as there are units in 7 x 10 ; but seven feet can no more be multiplied by ten feet than seven shillings by ten yards of silk.
When however given words imply a false proposition, there are two modes of proceeding, either to alter the words or to alter the meaning of the words. If a person should be so accustomed to talk of multiplying concrete quantities together that ho cannot avoid it, he must learn to define multiplication as the finding of a fourth pro.
portional to three concrete quantities, the first of which is a concrete unit. If this be the meaning of multiplication, then Rix yards and three yards can be multiplied together; for as one yard is to three yards so is six yards to eighteen yards, and eighteen yards is the product. But this product is a line, not an area The pertinacity with which some writers still persist in calling the product of two lines the area of a rectangle (not only as a practical rule of mensuration, in which it is a desirable mode of expression, but in matters of reasoning) is the result of a loug-continued habit formed in the first Instance by the study of the Greek writers. For though these do not confound the product with the area ; yet, on account of the deficiencies of their arithmetical system, they used the area instead of the product, and gave the names of spaces to the results of number's. Thus the product of two numbers was called plane, that of three equal numbers solid, that of two equal numbers a square, that of three equal numbers a cube, and the difference of two square numbers a gnomon. To these we may add the titles of polygonal, pyramidal, &c. numbers [N0MUERS, APPELLATIONS or], and others which it is needless to mention. All arithmetical propositions were made to take the form of geometrical ones ; thus to multiply two numbers was to form the rectangle of two given lines ; to divide one number by another was, Fircn the area of a rectangle and one of its sides, to find the other side. We have seen it stated that the word wapa,89Ah (parabola) was sometimes used for quotient, and, it was said, in Diophantus. We cannot find it there, though it may be used by the scholiast, whom we have not examined. Adrianue Romanus, in his ' Apologia pro Archimede,' says that the Greeks use rapa8ol4j for division as well as But most certainly the explanation of the meaning of a parabola, as applied to the well-known curve, comes from some such signification. The term parabola means a thing laid near to or by the side of another ; for comparison, for instance, as in the common word parable, or for any other purpose. Now in the conic seetiou in question the square on the ordinate being converted into a rectangle one of whose sides is the abscissa, the remaining side (being that which must be laid by the first side before the figure can be drawn, or the raeaBoa-6) is always of the same length. If modern writers had applied the term parabola to this remaining side, they would probably have called the curve an isoparabolic section ; but the Greeks, who called the curve in which a certain defect is always in the same pro portion to the whole by the simple name of defect (ellipse), and one haviug the same sort of excess by the simple name of excess (hyperbola), called the isoparabolic curve simply a parabola.