March i, 1900] 



NA TURE 



415 



while those on the other cut both in and out. This particu!?.r 

 property is not shown by the model as constructed, with the 

 present choice of co-ordinates. If, however, we had measured 

 entropy horizontally in the diagram, then the isentropics, being 

 vertical, might be tangent to cut through the steam-line. This 

 choice of co-ordinates has, however, seemed impossible for the 

 reasons previously given. 



We may if we wish discuss the question by a different method. 

 The lines drawn in Figs, i and 2 are all lines through the 

 critical point. In Fig. i the lines of constant pressure and 



Fig. 2. 



temperature are tangent to the broken line ; Fig. 2 shows the 

 same property. In Fig. i the line of constant entropy cuts the 

 broken line twice, but no other pair of lines has more than one 

 intersection. Fig. 2 does not, as drawn, show the same pro- 

 perty. In Fig. I, passing from the water-line around the critical 

 point in the homogeneous region to the steam-line, one cuts the 

 lines in the following order : water-line, pressure, temperature, 

 entropy, volume, pressure, temperature, entropy, steam-line. 

 Fig. 2 gives the same order, with the choice of co-ordinates, 

 which we have adopted, if we let the temperature lines always 



slope downward, as do the pressure lines. With this change 

 the two diagrams seem to agree, but otherwise their disagre- 

 ment seems hopeless. 



I shall be very glad to receive from any one any suggestion 

 which will help to remove the apparent disagreement between 

 the two diagrams, or so modify the model that it may more 

 completely represent the possible properties of actual bodies 

 than it now seems to do. W. B. BOYNTON. 



The University of California, 



Berkeley, Cal., U.S.A., February i. 



NO. 1583, VOL. 61] 



To Calculate a Simple Table of Logarithms. 



A YEAR ago Prof. Perry drew attention to a method by which 

 a schoolboy knowing how to extract square roots could, with 

 the help of squared paper, construct a table of logarithms 

 (Nature, February 23, 1899). 



It does not appear to be known that it is possible for a boy to 

 make a simple table of logarithms in a few minutes without 

 even knowing square root in arithmetic. 



Up to a few years ago the teaching of logarithms in schools 

 was generally deferred until they were required in trigonometry 

 for the solution of triangles, but the general introduction of 

 practical physics into secondary schools has resulted in the 

 teaching of logarithms to younger boys. 



The following method which I have introduced into several 

 Schools of Science in my district has been carefully tested in 

 classes of boys of about thirteen years of age with excellent 

 results. 



On a sheet of squared paper ruled in inches and tenths, plot 

 logarithms to base 2 : log 2=1 : log 4 = 2 : log 8 = 3 : log 16 = 4, 

 and draw a curve. 



It will be found convenient to arrange numbers from i to 16 on 

 a horizontal axis, taking i" as unit, and the logarithms on the 

 vertical axis, taking 3" as unit. 



From the curve read off the value of logjio, which will be 

 found to be approximately 3^. Let us assume that logoio is 

 exactly 3^. 



On any system of logarithms log 4 = 2 log 2 : log 8 = 3 log 2, 

 &c. Hence the curve obtained may be used to represent 



,.j . 



If ^ 



M ^-< 



3 -1 _jjp^ 



E 1 1 1 1 1 1 1 1 1 1 1 1 



The left-hand vertical column of figures in the diagram represent scale logs, 

 to base 2, and the right-hand column scale logs to base 10. 



logarithms to any base if the ordinates are measured on a 

 suitable scale. 



The scale used for measuring logs, to base 2 is a plain scale. 

 To construct a scale for measuring logs, to the base 10, write 

 log 10= I instead of 3^ ; and as this falls on the lOth line, the 

 distance from o to i can be at once divided into 10 parts, and 

 numbered 01 : 0*2, &c., the finer lines (not shown in the dia- 

 gram) giving the second decimal place. 



Having assumed that log2io = 3^, logio2 becomes '300 instead 

 of 301, so that the values from the curve are in error to the 

 extent of 1/300 ; but this is not greater than small errors due 

 to the freehand drawing of the curve and irregularities in the 

 ruling of the .-squared paper. Arthur Dufton. 



Sheffield, February 13. 



The publication of Mr. Dufton's method will, I think, serve 

 a useful purpose. It is a common exercise in schools to plot on 

 squared paper, numbers and their logarithms to the base 2 (see 

 Blaine's " Methods of Calculating," Spon), to give a general 

 notion of how the logarithm varies as the number varies ; but 

 I have never known it to be made a method of calculation. 

 Indeed, I do not think it right to give a boy the idea that he 

 may find log. 10 by interpolation between \of^ 8 and log. 16. 

 There is a specious appearance of accuracy due to the fact that 

 logjio is so nearly 3^ ; and Mr. Dufton heightens it by using 



