Page:The New International Encyclopædia 1st ed. v. 06.djvu/297

* riLXJVIUM. 253 DIMENSIONS. period. The term is little used by modern number of time-units: its 'dimensions' arc said writers, except perhaps in Cormany. L, ~-' ,, ,, DITttAN, JEREMI..H Lewis, D.D. (1831-81). *° ^« T °'" ^'^ ' "' therefore, on any systen. An -Vniericau Congregational theologian. He was of units, e.g. the C.G.S. system (q.v.), the bom in Bristol, R. 1., and died in Providenee, numerical value of a certain velocity is V, its where he had been professor of history and value on a system of units in which the unit of political economy in Brown University (his alma lengtli is ten times as largo as in the former mater) since 1804. He was an able preacher system would be V'/IO. Similarly, acceleration and orator, and is remembered by liis posthumous or the rate of change of velocity with reference publications, r/ie jT/ieisdV .li-<7u»ie)i( (1881), and. ., .., , ,. . /^V ™ Oral ious „nd f:ssays (ISSl). For his biography, t" the time has the dunensions (^^, j -^ T or consult Caroline Hazard (Boston, 1887). LT"'. Force is measured by the product of DIMANCHE, d.'-'maxsh', Monsieur. A mer- acceleration and mass; its dimensions are there- chant in .Moli&re's Don Jiinn, who unsuccessfully fore ilLT" - attempts to collect his bills. The terra is in use In the following table are given the dimen- in France as the synonj-m for a dun. sions of various mechanical quantities: DIKE (OF. dismr. Lat. decimiis, tenth, from Acceleration LT"' decern, ten). In the United States, a silver coin Force M LT"' whose value is ten cents — that is, one-tenth of a Pressure ML"'T"' dollar. Work 1 ML^T-' DIMENSION (Lat. dimensio, from dimetire, Enorgx / ■ to measure off. from di-, apart + metiri, to meas- Moment of force M L=T- ure). In geometry, a line, whether straight or If K and /* are written for the dimensions of curved, has one dimension, viz. length ; a plane electric and magnetic inductivity, the dimen- surface has two. length and breadth: and a sions of electric and magnetic quantities have solid has three, length, breadth, and thickness, the following values. In the first column are In algebra, the term dimension is applied in given the electro-static system of units ; in the much the same sense as degree, to express the second, the electro-magnetic. ( See Electricity. ) number of literal factors contained in a product, j., . ■ „uantitv T ^ f 4 T -' K ^ or T * M * u"* Thus x', xy. 2ab are said to be of two dimen- J^'ectuc quantity L,- M i^ K orL M-/i a-bc Electric current L^M*T 'K^orLni 5 T '^ * Mom: x>. x-y. -^ of three; and so on. In jj^g^^tj,. p^j^ ....L?M4t->} phvsieal measurements the power to which the Electric resistance LT"'/j unit (q.v.) of measure enters determines the Electro-motive force L* M* T '' K* dimension of the expression. The form of any ^^, l3M^T"°«i material bodv mav be described in terms of at _,, ^. .^ ^ ^_ _ -™ ", -i least three dimensions, hence the space in which Electric capacity. . .LK L ' T m these forms or figures exist is usually regarded The dimensions of any one physical quantity, as tridimensional. However, the possibility of however expressed, must be identical; and there- the existence of a space having more than three fore, choosing electric current dimensions has been discussed. The possibility L^ M- T~' K - = L- M* T~' /i~* or LT~' — of such space is conceivable and such an hypoth- -i — i t- t -Jm-i eds is of service to the mathematician in ex- /^ - K *. Hence /.k = L ^T *. plaining analytic phenomena. Since points, lines. This means that, although the dimensions of iind surfaces,' in general, generate bv their mo- neither electrical nor magnetic inductivity are tion lines, surfaces, and solids, respectively, it known in terms of length, mass, or tinie, the may be inferred that some movement of a figure product of the two has the dimensions L ^ T - > of three dimensions can generate a figure of four i.e. the same dimensions as the square root of dimensions. Analytically it is onlv necessary to the reciprocal of a velocity. imagine four parameters or coordinates belonging It should be noted that work and moment of to each point of four-dimensional space, or five force have the same dimensions ; but there is in five-dimensional space, and so on, in order a difference in this respect, in work the element to develop a svsteni of analysis as logical in it- of distance is in tlic direction of the force, while self as that of only three dimensions. Figures jn moment of force the element of distance is at to represent some features of four-dimensional right angles to the direction of the force. This bodies have been imagined. See Chasles; Chab- might be indicated by calling X and Y the di- ACTF.iiisTic : ,nnd Oeometkv. mensions of length in directions at right angles DIMENSIONS. All phvsieal quantities, such to each other: in which ca.se force has the dimen- as force, energj-, electric intensity, magnetic sions MXT"'; work, MX=T » ; moment of force, poles, etc., admit of mathematical expression in MXYT"^- terms of the elementary ideas of physics. Thus, In any equation connecting physical quanti- all mechanical quantities can be expressed in ties, it is evident that the dimensions of the terms of mass, length, and duration of time, quantities on the two sides of the equation must .Ml electrical and magnetic quantities can be ex- be identical; for a mass cannot equal a length, pressed in terms of mass, length, duration of etc. This fact is often useful cither in verifying time, and the 'inductivity' of matter for electric general conclusions or in predicating the Con- or magnetic forces. (See Er.ECTRTCfrY.) Thus, nection between various quantities. The subject velocity is defined as the limiting value of Ax to of dimensions is thoroughly treated in Daniel's A' where 4r is the distance traversed in the Text-Book of the Principles of Physics (London, time A'; consequently velocity involves the idea. 1894); Maxwell, Heat (London, 1891); Everett, of a number of units of length divided by a The C. G. S. System (London, 1875),