70 



NATURE 



[March i6, 1916 



(2) The shorter measures of distance, such as the foot, 

 the yard, and the pace. (3) The longer measures of 

 distance, including- the stadium, the mile, the para- 

 sang, the schoenos, the league, the hour's march, 

 and the day's march. (4) .Measures of length used in 

 connection with the calculation of land areas, of which 

 the English representatives are the perch, the chain, 

 and the furlong. 



As regards the first of these classes of measures, it 

 is generally accepted that they were, from the earliest 

 times, based on the proportions of the human body, so 

 that every man had his own scale to which he could 

 work. 



The palm is the width across the open hand at the 

 base of the fingers ; the cubit is the length of the arm 

 from the elbow to the end of the middle finger; and 

 the fathom the length of the outstretched arms. 

 There is no fixed relationship between these units. 



There is no record as to when an attempt was first 

 made to combine the measures in a standard scale, 

 but it was probably at an early period, as it must 

 have been found inconvenient for workers on the 

 same building, for example, to use different lengths 

 of palms and cubits, and, when a standard was fixed, 

 it may have been some such scale as the following : — 



I digit = 07375 EngUsh inch 

 4 digits = I palm = 2-95 „ inches 



6 palms = I cubit = 1770 „ „ 



The cubit of this scale may be called the " cubit of 

 a man," to distinguish it from other cubits, which 

 will be described hereafter. 



There is nothing to show when the foot was added 

 to the units of the mechanic's scale, but when this was 

 done it was assumed to be equal to four palms, or 

 two-thirds of a cubit. 



The third class of measures of length is the most 

 'important, and the history of these is of particular 

 interest, as they appear to have started in a state of 

 perfection, and to have been first used by a people 

 who possessed a high degree of astronomical and 

 mathematical knowledge, who were acquainted with 

 the form of the earth, and were able to carry out 

 geodetical measurements. There can be no doubt 

 that they are based on the angular division of the 

 circle, and on the application of this division to 

 terrestrial measurements. 



The unit of angular measurement is the angle of 

 an equilateral triangle, and this angle was divided 

 by the ancient geometricians, for purposes of cal- 

 culation, into 60°, the best number possible, as 

 60 = 3x4x5. Following the same principle, each 

 degree was divided into 60 minutes, and each minute 

 into 60 seconds. As the circle contains six times the 

 angle of an equilateral triangle the circle was divided 

 into 360°. Thjs division of the circle, although so 

 ancient that its origin is unknown, has never been 

 improved upon, and is still in use by all nations. 

 An attempt on the part of certain French mathe- 

 maticians to substitute a division of the circle into 

 400°, on account of the supposed advantages of the 

 decimal system, has proved a failure. 

 ■ The manner in which the division of the circle into 

 360° was used by the ancients to determine the unit for 

 terrestrial measures of distance was as follows. If a 

 circle be described cutting the equator of the earth at 

 right angles, and passing through the north and south 

 ]X)les. its circumference in angular measurement is 

 equal to :^6o° x 60' = 2 1.600', and the length of i minute, 

 measured on the surface of the globe, was taken as 

 the unit, which is called a geographical mile at the 

 present time. If the earth was a perfect sphere, every 

 geojrraphical mile would be of the same length, but, 

 as the polar diameter is less than the equatorial 

 diameter in the proportion of 7900 to 7926, the length 



NO. 2420, VOL. 97] 



of the geographical mile, measured on the meridian, 

 is not the same in all latitudes, but increases in length 

 from 6046 English feet at the equator to 6108 English 

 feet at the poles. Whether the ancient astronomers 

 were acquainted with this irregularity in the figure 

 of the earth it is not possible to say, "but it is certain 

 that the value at which they fixed it must have been 

 close to the actual mean value as determined by 

 modern astronomers, which may be taken as about 

 6075 English feet. The Greek stadion (the same as 

 the Roman stadium), which was one-tenth of the 

 geographical mile, was 600 Greek feet In length, and 

 the Greek foot was about I2'i5 of our present English 

 inches. 



The next step taken appears to have been with the 

 view of assimilating the subdivisions of the geo- 

 graphical mile with the cubit, and it was not easy 

 to do this, as the cubit of a man has no necessary 

 connection with a geographical mile. The diflficulty 

 appears to have been solved by the invention of two 

 new cubits, of which the smaller was very nearly 

 equal to the cubit of a man, and was contained 4000 

 times in the geographical mile. This, for the sake 

 of distinction, may be called the geographical cubit. 

 The second cubit, afterwards known as the Babylonian 

 Royal cubit, was longer, and was contained 3600 times 

 in the geographical mile. According to Herodotus, 

 this second cubit was three digits longer than the 

 other cubit. On these two cubits there appear to have 

 been based two different divisions of the geographical 

 mile, one in accordance with a decimal, and the other 

 with a sexagesimal system of calculation, but there 

 is, so far as I know, no ancient record of these scales, 

 and the following attempt to compose them is founded 

 on Inferences, drawn from the Babylonian, Greek, and 

 Roman measures, all of which, there can be little 

 doubt, came from the same origin. 



The first based on the geographical cubit, which 

 was rather longer than the average cubit of a man, 

 Is as follows :— 



I digit = 0729 English inch 



25 digits = I geographical cubit =18-225 „ inches 

 100 „ = I fathom = 6075 ,, feet 



100 fathoms = I stadion = 607*5 „ ,, 



10 stadia = i geographical mile = 6075 ,, „ 



The second, or sexagesimal scale, based on the 

 Babylonian Royal cubit, appears to have been as 

 follows : — - 



I digit = o 723 English inch 



28 digits = I Royal cubit = 20*25 >> inches 



60 cubits = I plethron =ior25 „ feet 



60 plethra= I geographical mile = 6075 ,, ,, 



The ancient Egyptian measures of length, although 

 evidently derived from the same origin as the Baby- 

 lonian, differ from these In some respects. The most 

 important smaller unit was a cubit usually known as 

 the Egyptian Royal cubit, which was divided into 

 seven palms, each palm of four digits. The approxi- 

 mate length of the Egyptian Royal cubit Is well 

 known, as a number of cubit scales have been found 

 which give a mean length of 20-65 English inches, and 

 an examination of the monuments of Egvpt shows 

 that this cubit was used for building purposes from 

 ancient times. 



It Is matter of controversy from whence the Greeks 

 derived their measures of length, whether from Egypt 

 or Babylonia ; but the latter appears more probable, 

 as their principal measure of distance, the stadion, 

 was equal to one-tenth of a geoq-raphical mile of 6075 

 English feet, and this was divided Into 6 plethra, each 

 of 100 Greek feet. The Greek scale appears to have 

 been as follows : — 



