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MENSURA.

Sense, signifies the application of number to quan­tity ; or, to speak more specifically, the comparison of'different quantities by means of the standard of number. So long as we regard quantity apart from number, we can only compare two quantities by the test of coincidence, by which we ascertain whether they are equal or unequal, and, if the latter, which of the two is the greater ; as, for in­stance, in the case of two lines. The next step is the comparison of one magnitude with certain de­finite parts, or multiples, of the other, its half or double, third or triple, and so forth. The last step, by which we attain to a complete method of expressing magnitude numerically, is the choice of some fixed magnitude, or unit, with which we may compare all other magnitudes of the same kind, so as to ascertain what multiple, part, or parts of the unit each of them is, if they are commen­surable, and, if not, as nearly as we please. Thus the unit, in itself, or in its parts, forms a Measure of all magnitudes of the same kind as itself. A set of fixed measures, one for each kind of quantity, with their subdivisions, forms a Metrical System.

The notions which lie at the foundation of ma­thematical and mechanical science determine of themselves the foundation of every metrical system. Those notions are Extension and Force ; the former in its various kinds, the line, the surface, the solid, and the angle; the latter in that manifestation of it which we call weight. Now, since extension, whether linear, superficial, or solid, can be esti­mated by means of one straight line ; or by means of two straight lines which form a fixed angle with one another, and which, together with two other lines drawn parallel to them, enclose a surface ; or by means of three straight lines, the planes passing through which form a fixed solid angle, and, to­gether with three other planes drawn parallel to them, form a solid: —it follows that all these three kinds of magnitude may be estimated numerically by fixing upon units which are respectively a straight line, a parallelogram having two adjacent sides and an angle fixed, and a parallelepiped having three adjacent edges and an angle fixed ; or, simplifying the two latter cases by making the iixed sides equal and the fixed angles right angles, the units are (1) a straight line of fixed length, (2) the square of which that straight line is a side, and (3) the cube of which that Line is the edge. Thus we obtain a metrical system for length, surface, tmd capacity.

For the measurement of angular magnitude, or, which is the same thing, of distance reckoned along the circumference of a circle, one unit is sufficient, namely, a fixed angle, which will exactly measure the sum of four right angles, or a fixed arc of a iixed circle, xvhich will exactly measure the cir­cumference of the circle. Thus we obtain a me­trical system for all angular magnitudes, including Time.

Again, with respect to Force, of which the test is weight, since all forces may be compared, either directly, or through the calculation of the velocities which they produce, with the force of gravity. There are two ways of estimating weight. Either its measure may be deduced from the measure of capacity ; for, as the weight of a body depends on the quantity of matter in a given space, estimated by the effect which the force of gravity exerts upon it, we may take the quantity of a fixed kind of matter (water for example) which will exactly fill

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MENSURA.

the unit of capacity, as the unit of weight. Or we may take a bulk of any substance, without, measuring it, as the unit of weight. In the latter case it is evident that, by measuring the solid con­tent either of the unit of weight, or of an equal weight of some other substance, we might derive from our system of weights a system of measures, first, of capacity, and thence of surface, and thence of linear distance ; just as by the opposite process we pass from the line to the surface, thence to capacity, and thence to weight.

The statement of these elementary principles, in as brief a form as is consistent with clearness, has appeared necessary, in order to the complete un­derstanding of the metrical systems of the Greeks and Romans, the explanation of which is the object of this article.

I. Origin of Measures.

\. Of Length. — The first step in the construction of a metrical system is obviously that of fixing upon the unit of length ; and nature itself suggests the choice, for this purpose, of some familiar object, of nearly uniform length, and which is constantly at hand to be referred to. These conditions are fulfilled by various parts of the human body ; from which accordingly we find that not only the unit of length, but all the measures of length, except those which are too small or too large to be mea­sured by parts of the body, are derived in every metrical system, except the latest formed of all, the modern French system, which is founded on the measurement of the earth. In support of tlio general statement now made we have, besides the-antecedent argument from the nature of the case, the testimony of all writers, the names of the measures, and the general agreement of their lengths with the parts of the body whose namea they bear. (Horn. II. vi. 319, xv. 678, Od. xi 310 ; Vitruv. iii. 1. § 2 — 9, with Schneider's Notes; Hero, Geom. in Anal. Graec. Paris, 1688, vol. i. pp. 308—315, 388 ; Diog. Laert. ix. 51 ; Ukert, Geog. d. Griech. u. Rom. vol. i. pt. 2, p. 54.) The chief of such measures, with their Greek and Roman names, are the following : tho breadth of a finger (SdKrvXos, digitus) or thumb (pollex} ; the breadth of the hand, or palm (wa-XaiffTi}, palmus) ; the span, that is, the distance from the tip of the thumb to the tip of the little finger, when spread out as wide as possible (o"rriOaijL7]') * ; the length of the foot (irovs, pes) ; the cubit, or distance from the elbow to the tip of the middle finger (irrJXVSi) cubitus) ; a step (fi?nm, gradus) • a double step, or pace (passus) ; and the distance from extremity to extremity of the out­stretched arms (opyvid). With reference to the last two measures, it will be observed that the Romans derived them from the legs, the Greeks from the arms, the passus being one foot shorter than the opyvid of the other, and the former (5 feet) belonging to the decimal system, the latter (6 feet) to the duodecimal. The higher measures of length will be referred to presently. Comp. Pol­lux, ii. 157, 158 ; who also mentions some less important measures ; namely, the §0x^77 or 8a/c-or ^oopov, which was the same as the f] ; the 6p(?o8%)oz', or the length of the

* This measure was not in the Ptoman system, When they wished to express the Greek span, the proper word was do'drans, that is, three quar­ters (of the foot).

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