September 22, 1892] 



NATURE 



491 



his chart, now of universal use in navigation. In it every 

 island, every bay, every cape, every coast-line, if not 

 extending over more than two or three degrees of longi- 

 tude, or farther north and south than a distance equal 

 to two or three degrees of longitude, is shown very 

 approximately in its true shape : rigorously so if it 

 extends over distances equal only to an infinitesimal 

 difference of longitude. The angle between any two 

 intersecting lines on the surface of the globe is repro- 

 duced rigorously without change in the corresponding 

 angle on the chart. 



Mercator's chart may be imagined as being made by 

 coating the whole surface of a globe with a thin inex- 

 tensible sheet of matter — sheet india-rubber for example 

 (for simplicity, however, imagined to be perfectly ex- 

 tensible but inelastic) — cutting away two polar circles to 

 be omitted from the chart ; cutting the sheet through 

 along a meridian, that of i8o" longitude from Greenwich 

 for example, stretching the sheet everywhere except 

 along the equator so as to make all the circles of latitude 

 equal in length to the circumference of the equator, and 

 stretching the sheet in the direction of the meridian in 

 the same ratio as the ratio in which the circles of latitude 

 are stretched, while keeping at right angles the inter- 

 sections between the meridians and the parallels. The 

 sheet thus altered may be laid out flat or rolled up, as a 

 paper chart. 



What I call a generalized Mercator's chart for a body 

 of any shape spherical or non-spherical, is a flat sheet 

 showing for any intersecting lines that can be drawn on 

 a part of the surface of the body, corresponding lines 

 which intersect at the same angles. One Mercator chart 

 of finite dimensions can only represents part of the com- 

 plete surface of a finite body, if the body be simply 

 continuous ; that is to say, if it has no hole or tunnel 

 through it. The whole surface of an anchor ring can 

 obviously be mercatorized on one chart. It is easily seen, 

 for the case of the globe, that two charts suffice to mer- 

 catorize the whole surface ; and it will be proved presently 

 that three charts suffice for any simply continuous closed 

 surface, however extremely it may deviate from the 

 spherical form. 



In " Liouville's Journal" for 1847, its editor, Liouville, 

 gave an analytical investigation, according to which, if 

 the equation of any surface whatever is given, a set of 

 lines drawn on it can be found to fulfil the condition that 

 the surface can be divided into infinitesimal squares by 

 these lines and the set of lines on the surface which cut 

 them at right angles. Now it is clear that if we have any 

 portion of a curved surface thus divided into infinitesimal 

 square allotments, that is to say, divided into infinitesimal 

 squares, with the corners of four squares together, all 

 through it, we can alter all these squares to one size and lay 

 them down on a flat surface with each in contact with its 

 four original neighbours ; and thus the supposed portion of 

 surface is mercatorized. Except for the case of a figure of 

 revolution, or an ellipsoid, or virtually equivalent cases, 

 Liouville's differential equations are of a very intractable 

 kind. I have only recently noticed that .we can solve the 

 problem graphically (with any accuracy desired if the 

 problem were a practical problem, which it is not) by aid 

 of a voltmeter and a voltaic battery, or other means of 

 producing electric currents, as follows : — 



1. Construct the surface to be mercatorized in thin 

 sheet metal of uniform thickness throughout. By thin I 

 mean that the thickness is to be a small fraction of the 

 smallest radius of curvature of any part of the surface. 



2. Choose any two points of the surface, N, S, and 

 apply the electrodes of a battery to it at these points. 



3. By aid of movable electrodes of the voltmeter, trace 

 an equipotential line, E, as close as may be around one 

 electrode, and another equipotential line, F, as near as 

 may be around the other electrode. Between these two 

 equipotentials, E, F, trace a large number, «, of equi- 



NO. I I 95, VOL. 46] 



different equipotentials. Divide any one of the equi- 

 potentials into n equal parts ; and through the divisional 

 points draw lines cutting the whole series of equipoten- 

 tials at right angles. These transverse lines and the 

 equipotentials divide the whole surface between E and F 

 into infinitesimal squares (Maxwell, " Electricity and 

 Magnetism," § 651). 



4. Alter all the squares to one size and place them 

 together, as explained above. Thus we have a Mercator 

 chart of the whole surface between E and F. 



N and S of our generalization correspond to the north 

 and south poles of Mercator's chart of the world ; and 

 our generalized rule shows that a chart fulfilling the 

 essential principle of similarity realized by Mercator may 

 be constructed for a spherical surface by choosing for 

 N, S any two points not necessarily the poles at the ex- 

 tremities of a diameter. If the points N, S are infinitely 

 near one another, the resulting Mercator chart for the 

 case of a spherical surface, is the stereographic projection 

 of the surface on the tangent plane at the opposite end 

 of the diameter through the point, C, midway between 

 N and S. In this case the equipotentials and the stream- 

 lines are circles on the spherical surface cutting N S at 

 right angles, and touching it, respectively. 



For a spherical or any other surface we may mercatorize 

 any rectangular portion of it, A B C D, bounded by four 

 curves, AB, BC, CD, DA, cutting one another at right 

 angles as follows. Cut this part out of the complete 

 metallic sheet ; to two of its opposite edges, A B, D C, 

 for instance, fix infinitely conductive borders. Apply the 

 electrodes of a voltaic battery to these borders, and trace 

 n equidifferent equipotential lines between AB and DC. 

 Divide any one of these equipotentials into « equal parts, 

 and through the divisional points draw curves cutting 

 perpendicularly the whole series of equipotentials. These 

 curves and the equipotentials divide the whole area into 

 infinitesimal squares. Equalize the squares and lay them 

 together on the flat as above. 



If we have no mathematical instruments by which we 

 can draw a system of curves at right angles to a system 

 already drawn, we may dispense with mathematical 

 instruments altogether, and complete the problem of 

 dividing into squares by electrical instruments as fol- 

 lows : Remove the conducting borders from AB, DC ; 

 apply infinitely conductive borders to AD and BC, apply 

 electrodes to these conducting borders, and as before 

 draw 11 equidifferent equipotentials. This second set of 

 equipotentials, and the first set, divide the whole area 

 into squares, Kelvin. 



THE ACTIVE ALBUMEN IN PLANTS. ^ 



ONE of the most important chemical functions of 

 j plant-cells is that synthesis of albuminous matter 



j which serves for the formation of protoplasm. Tht living 

 protoplasm, however, is composed of proteids entirely 

 different from the ordinary soluble proteids, as well as 

 from the proteids of dead protoplasm. In other words, if 

 living protoplasm dies, the albuminous constituents 

 change their chemical character. We observe that in the 

 living state a faculty of antoxidation (respiration) exists, 

 which is wanting in the dead condition ; and Pfliiger, in 

 1875, drew from this the conclusion that in protoplasm 

 the chemical constitution of the living proteids changes 

 at the moment of death. 



Various other considerations force us to accept this 

 logical conclusion. Chemical changes readily occur in 

 all those organic compounds that are of a labile char- 

 acter. There exist so-called labile atom- constellations 

 that are in lively motion, and are thus prone to undergo 

 change, the atoms falling into new arrangements which 



I This paper was read before the Liege meeting of the Interrational Con- 

 gress of Physiologists, of whose proceedings we gave some account lastweek. 



