May 24, 1877] 



NATURE 



67 



The radial bars are of equal length — I employ the word 

 "length " for brevity, to denote the distance between the 

 pivots, the links, of course, may be of any length or 

 shape, — and the distance between the pivots or the tra- 

 versing link is such that when the radial bars a' e parallel 

 the line joining those pivots is perpendicular to the radial 

 bars. The tracing-point is situate half-way between the 

 pivots on the traversing piece. The curve described by 

 the tracer is, if the apparatus does not deviate much from 

 its mean position, approximately a straight line. The 

 reason of this is that the circles described by the ex- 

 tremities of the radial bars have their concavities turned 

 in opposite directions, and the tracer being half-way be- 

 tween, describes a curve which is concave neither one 

 way nor the other, and is therefore a straight line. The 

 curve is not, however, accurately straight, for if I allow 

 the tracer to describe the whole path it is capable of 

 describing, it will, when it gets some distance from its 

 mean position, deviate considerably from the straight line, 

 and will be found to describe a figure 8, the portions at 

 the crossing being nearly straight. We know that they 

 are not quite straight, because it is impossible to have 

 such a curve partly straight and partly curved. 



For many purposes the straight line described by 

 Watt's apparatus is sufficiently accurate, but if we require 

 an exact one it will, of course, not do, and we must try 

 again. Now it is capable of proof that it is impossible to 

 solve the problem with three moving links ; closer ap- 

 proximations to the truth than that given by Watt can be 

 obtained, but still not actual truth. 



I have here some examples of these closer approxima - 

 tions. The first of these, shown in Fig. 3, is due to 

 Richard Roberts of Manchester. 



The radial bars are of equal length, the distance be- 

 tween the fixed pivots is twice that of the pivots on the 

 traversing piece, and the tracer is situate on the traversing 

 piece, at a distance from the pivots on it equal to the 

 lengths of the radial bars. The tracer in consequence 

 coincides with the straight line joining the fixed pivots at 

 those pivots and half-way between them. It does not, 

 however, coincide at any other point, but deviates very 

 slightly between the fixed pivots. The path described by 

 the tracer when it passes the pivots, altogether deviates 

 from the straight line. 



The other apparatus was invented by Prof. Tchebicheff 

 of St. Petersburg. It is shown in Fig. 4. The radial 

 bars are equal in length, being each in my little model 

 five inches long. The distance between the fixed pivots 

 must then be four inches, and the distance between the 

 pivots or the traversing bar two inches. The tracer is 

 taken half-way between these last. If now we draw a 

 straight line — 1 had forgotten that we cannot do that yet, 

 well, if we draw a straight line, popularly so called — 

 through the tracer in its mean position as shown in the 

 figure, parallel to that forming the fixed pivots, it will be 

 found that the tracer will coincide with that line at the 

 points where verticals through the fixed pivots cut it as 

 well as at the mean position, but, as in the case of 

 Roberts's parallel motion, it coincides nowhere else, 

 though its deviation is very small as long as it remains 

 between the verticals. 



We have failed then with three links, and we must go on 

 to the next case, a five-link motion — for you will observe 

 that we must have an odd number of links if we want an 

 apparatus describing definite curves. Can we solve the 

 problem with five ? Well, we can, but this was not the 

 first accurate parallel motion discovered, and we must 

 give the first inventor his due (although he did not find 

 the simplest way), and proceed in strict chronological 

 order. 



In 1864, eighty years after Watt's discovery, the pro- 

 blem was first solved by M. Peaucellier, an officer of 

 Engineers in the French army. His discovery was not at 

 first estimated at its true value, fell almost into oblivion, 

 and was rediscovered by a Russian student named Lipkin, 

 who got a substantial reward from the Russian Govern- 

 ment for his supposed originality. However, M. Peau- 

 cellier's merit has at last been recognised, and he has 

 been awarded the great mechtnical prize of the Institute 

 of France, the " Prix Montyon." 



M. Peaucellier's apparatus is shown in Fig. 5. It has, 

 as you see, seven pieces or links. There are first of all 

 two long links of equal length. These are both pivoted 

 at the same fixed point ; their other extremities are 

 pivoted to opposite angles of a rhombus composed of four 

 equal shorter links. The portion of the apparatus I have 

 thus far described, considered apart from the fi.xed bnse, 

 is a linkage termed a " Peaucellier cell." We then take 

 an ex/ra link, and pivot it to a fixed point whose distance 

 from the first fixed point, that to vvhich the cell is pivoted, 

 is the same as the length of the extra link ; the ottier end 

 of the extra link is then pivoted to one of the free angles 

 of the rhombus ; the othtr free angle of the rhombus has 

 a pencil at its pivot. That pencil will accurately describe 

 a straight line. 



I must now indulge in a little simple geometry. It is 

 absolutely necessary that I should do so in order that you 

 may understand the principle of our apparatus. 

 {To be continued?) 



FOSSIL FLORAS AND GLACIAL PERIODS 



A RECENT notice in Nature (vol. xiv.p. 336) of certain 

 -'*- inferences of Prof. Heer in connection with the Arctic 

 fossil plants obtained by the Swedish Expeditions of 1870 

 and 1872, suggests some thoughts on the relations of fossil 

 plants to climate, wtiich, though I have discussed them 

 elsewhere, deserve to have attention again directed to 

 them. In my Bakerian Lecture before the Royal Society 

 in 1870, and in my " Report on the Pre carboniferous 

 Flora of Canada," published by the Canadian Survey 

 in 1871, I deduced from the generalisations of Prof. 

 James Hall as to the growth of the American Continent 

 from the north-east, in connection with the distribution of 

 the fossil plants of the Upper Silurian, Erian, and Carbo- 

 niferous systems, the conclusion that these assemblages 

 of plan's entered North America from the north-east, and 

 propagated themselves southward and westward. Prof. 

 Asa Gray had, as early as 1867, stated similar conclusions 

 with reference to the modern floras of America and 

 Eastern Asia, and has more recently extended them to 

 the Tertiary floras on the evidence of Heer and Les- 

 quereux. ' 



The further conclusion that all the old floras appeared 

 suddenly and abruptly in the temperate regions, and with 

 a great number of species, I have illustrated in the Re- 

 port above referred to, as far as regards the Paleozoic 

 plants, and have referred to the evidence of it in the case 

 of the Cretaceous and Tertiary floras in my address to 

 the American Association in 1 87 J. 



With regard to the succession of these floras, it is true 

 that it has been the fashion with certain European palae- 

 ontologists to regard our rich Devonian or Erian flora 

 ' Adtiress to the American Association, 1872. 



