October 5, 1883.] 



SCIENCE. 



483 



be the tracing-point and tbe pencil of a pentagiaph. 

 ■you may with tlie fiisl point draw any figure you 

 please: there will he a corresponding figure drawn by 

 the second point, — for a good pcntagraph, a copy 

 on a scale different, it may he; for a badly adjusted 

 pentagraph, a distorted copy; hut the one figure will 

 always be a sort of copy of the first, so that to each 

 point of the one figure there will correspond a point 

 in the other figure. 



In the case above referred to, where one point rep- 

 resents the value x+ 1// of the imaginary variable, and 

 llieolher the value X-l-ii'of some function, ^ [x+iy), 

 of that variable, there is a remarkable relation be- 

 tween the two figures : this is the relation of orlho- 

 morphic projection, the same which presents itself 

 between a portion of the earth's surface and the rep- 

 resentation thereof by a map on the stereograpbic 

 projection or on Jlercator's projection; viz., any in- 

 definitely small area of the one figure is represented 

 in the other figure by an indefinitely small area of 

 the same shape. There will possibly be for differ- 

 ent parts of the figure great variations of scale, but 

 the shape will be unaltered. If for the one area 

 the boundary is a circle, then for the other area the 

 boundary will be a circle: if for one it is an equilat- 

 eral triangle, then for the other it [will be an equi- 

 lateral triangle. 



I have been speaking of an imaginary variable 

 {x+iy), and of a function, 9(x+iy)—X+iT, of that 

 variable: but the theory may equally well be stated 

 in regard to a plane curve: in fact, the x+iy and 

 the X-|-iF are two imaginary variables connected 

 by an equation. Say their values are u and v, con- 

 nected by an equation, F (u, v) — 0: then, regard- 

 ing u, t, as the co-ordinates of a point in piano, this 

 will be a point on the curve represented by the equa- 

 tion. The curve, in the widest sense of the expres- 

 sion, is the whole series of points, real or imaginary. 



the co-ordinates of which satisfy the equation; and 

 these are exhibited by the foregoing corresponding 

 figures in two planes. But, in the ordinary sense, the 

 curve is the series of real points, with co-ordinates «, 

 V, which satisfy lheC(iuation. 



In geometry it is the curve, whether defined by means 

 of its equation or in any other manner, which is the 

 subject for coutemplation aiul study. But we also 

 use the curve as a representation of its equation; 

 that is, of the relation existing between two magni- 

 tudes, X, y, which are taken as the co-ordinates of a 

 point on the curve. Such employment of a curve 

 fur all sorts of purposes — the fluctuations of the 

 barometer, the Cambridge boat-races, or the funds — 

 is familiar to most of you. It is in like manner con- 

 venient in analysis, for exhibiting the relations be- 

 tween any three magnitudes, x, y, z, to regard them 

 as the co-ordinates of a point in space; and, on the 

 like ground, we should at least wish to regard any 

 four or more magnitudes as the co-ordinates of a 

 point in space of a corresponding number of dimen- 

 sions. Starting with the hypothesis of such a space, 

 and of points therein, each determined by means of 

 its co-ordinates, it is found possible to establish a 

 system of n-dimensional geometry analogous in every 

 respect to our two- and three-dimensional geometries, 

 and to a very considerable extent serving to exhibit 

 the relations of the variables. 



It is to be borne in mind that the sp.ace, whatever 

 its dimensionality may be, must always be regarded 

 as an imaginary or complex space, such as the two- or 

 threc-dimeusional space of ordinary geometry. The 

 advantages of the representation would otherwise 

 altogether fail to be obtained. 



I omit some farther developments in regard to 

 geometry, and all that I have written as to tlie con- 

 nection of mathematics with the notion of time. 

 (To be continued.) 



INTELLiaENCE FROM AMERICAN SCIENTIFIC STATIONS. 



STATE INSTITUTIONS. 



Illinois state laboratory of natural history. Normal, 111. 



Experiments with diseased caterjiillars. — Prof. S. 

 A. Forbes is making a .special study of ' schlaffsucht,' 

 or some very similar disease, among our native cat- 

 erpillars. He has so far proven that the disease is 

 characterized by an enormous development of bac- 

 teria in the alimentary canal, the same forms appear- 

 ing in the blood before death; that it is contagious 

 by way of the food ingested; that the characteristic 

 bacteria may be easily and rapidly cultivated in ster- 

 ilized beef-broth; and that caterpillars whose food 

 hiis been moistened with this infected broth, speedily 

 show the bacteria in the alimentary canal, and, later, 

 in the blood, and soon all die of the disease. Other 

 caterpillars of the same lot, receiving the same treat- 

 ment, except that the food is moistened with distilled 

 water instead of the infected broth, remain unaf- 



fected. These bacteria are likewise cultivable in 

 vegetable infusions, but multiply there less freely. 



Every step of the investigation is fortified by 

 st:iined and mounted preparations, which are being 

 submitted to cryptogamists. It h.as already been 

 determined that the bacterium infesting a brood of 

 Datana ministra in his breeding-cages is identical 

 with the Micrococcus bombycis of the silk-worm; 

 the form, measurements, modes of aggregation, and 

 behavior to reagents, of the two, being the same. 

 Dalana Angusii, feeding upon walnut, >vas also occa- 

 sionally infested by this M. bombycis, but much 

 more commonly by a spherical species, probably un- 

 described. 



In the cabbage-worm (Pieris rapac) occurs still 

 another species of Micrococcus, very minute (5/tfin 

 diameter), globular, and usually either single or in 

 pairs. This is far the most virulent of the insect 

 affections, which is being studied by Forbes, — the 



