558 



NA TURE 



{April 14, 1887 



Association, piinled in the twelfth Annual Report, 1886, I feel 

 impelled, on at length receiving to-day, at this frontier outpost 

 of scientific civilisation, a copy, long ordered, of the Associa- 

 tion's " Plane Geometry," Part 2, to say a few words suggested 

 by it, fa\'0uring accord in these fundamental matters. 



The very first definition and first theorem show the glaring 

 need in English for a word which the Germans have in Strecke. 

 Such a word, meaning a piece of a straight line, is needed in 

 the first definition, the definition of a circle, for all straight 

 lines are infinite in size, and radii are pieces of straight lines, 

 and not whole straight lines. This is unconsciously recognised, 

 even in the first theorem, where for "piece of a straight line" 

 the undefined word distance is used, inappropriate because of its 

 association with ideas of measurement by a unit and length, and 

 because of its different and confusing use in the phrase "shortest 

 distance." 



In the demonstration of this first theorem, "straight line" is 

 used in its proper sense, though just before, in the first defini- 

 tion, it was bunglingly used in the sense here given to distance. 

 For the part of a straight line between two definite points I have 

 long used the word "sect," which, carried over to the sphere 

 with the meaning part of the spherical line less than half, gives 

 the key to two-dimensional spherics. 



And this suggests another objection to the same first defini- 

 tion. It says a circle is a plane figure. Now one cannot even 

 think of spherics without seeing how immeasurably better it is 

 to define a circle as a curve. It will be so defined as soon as the 

 student reaches analytics, sd why have him learn something 

 only to unlearn ? 



In the fourth definition we have an over-used word, " conju- 

 gate." Two arcs which together make a circle should be called 

 ex/lemciilal. Exjilement is a natural third to complement and 

 supplement. Again converse is a term of logic, and does not 

 mean wliat it is here used to mean, tliat is, inverse. 



In the introductory remarks to the fourth book I think it is a 

 mistake to call hunger, love, courage, talent, wisdom, magni- 

 tudes. 



A magnitude is whatever can be added to itself, so a^ to 

 double. The very first sentence says : "In this book the sub- 

 jects of the propositions . . . are magnitudes in general ; " but 

 the whole treatment is founded upon tuultiples, and is only 

 applicable where multiples can be made. Not only must we 

 have an exact criterion of equality, we must be able to add with- 

 out shrinkage. 



A little farther on we meet the absurd statement, "Funda- 

 mentally, number is counting." 



Now we know that counting is establishing a one-to-one corre- 

 spondence between the individuals of an aggregate and of a 

 standard group which was primarily the fingers. But a number 

 is fundamentally a picture of an aggregate which for all counting 

 purposes is as good as the aggregate itself — a picture consisting 

 of a mark for each distinct individual in the aggregate, as III. ; 

 and then secondarily a symbol for that picture, as 3. 



It is questionable whether Book IV. Part I, has any valid 

 excuse for existing. Proportion for commensurable magnitudes 

 neither calls for nor warrants treatment by multiples. Sande- 

 man, in the preface to his " Pelicotetics," speaks of "the phe- 

 nomenon of incommensurability, through which alone arises any 

 need of ratio, either the thing or the name." Euclid's mar- 

 vellously elegant treatment of proportion is only admirable 

 because of the difficulty it so deftly overcomes. To use it on 

 commensurables is to use a Galling gun on a plucked chicken. 

 The illustration given under Definiii m 4 of this Part I (which 

 definition needs the word commensurable inserted in it), "4half- 

 crowns = 5 florins," reminds one how badly England needs a 

 decimal system of coinage, weights, and measures. No light is 

 thrown on the compounding of ratios, but the error of A. J. 

 Ellis is avoided. He says : "The ratio of B to A means the 

 order in which the multiples of B are distributed among those 

 of A." 



These are points suggested in first turning the leaves of a 

 new book of most gratifying soundness. May it ward off from 

 England the misfortune America now sutlers, in that our most 

 p ->pular book on geometry makes the fundamental blunder of 

 basing the treatment of parallels on direction, uses in its proofs 

 the stultifying formula, " a straight line is the shortest distance 

 between two points," and from one end to the other makes us 

 wish for an American Association for the Improvement of Geo- 

 metrical Teaching. George Bruce Halsted 



University of Texas, Austin, Texas, March 3 



The Svastika as both Sun and Fire Symbol 

 The late Prof. Dr. Worsaae ("Industrial Arts of Old De 



mark ") claims the ring-cross 



e 



as he terms it, as 



symbol, and a small cup-shaped hollow for the moon ; botli 

 these he places as belonging to the later Stone Age of Scandi 

 navia, and, apparently, the only recognised emblems of thai 

 period. He observes, in one place : " How many hundred 

 years, or, indeed, how many thousand years, before the Chris- 

 tian era the earlier Stone Age began, it is impossible to say." 

 The same writer places amongst tlie emblems of the late.- 



Bronze Age the wheel-cross 



(the chariot wheel of the 



sun ?). To thii day, both in Denmark, Holland, and in parts 

 of Germany, a wheel is frequently placed on the roof of a stable 

 or other building, which is thus deemed protected from fire, 

 especially if a stork can be induced to make its nest upon the 

 wheel. The stork, owing to his red legs, was not inaptly con- 

 sidered an emblem of fire ; he was also the herald of summer — 

 he brought light and warmth. The Moqui symbol for the sun (as 

 described by Dr. Dryer in Nature, Feb. 10, p. 345) exists also 

 on articles classed by Prof. Worsaae a; belonging to the later 

 Bronze Age in .Scandinavia, with the exception of the three 

 marks of which he speaks, as indicating the eyes and mouth of 

 a face. 



According to Hyde (" Persian Religion," p. 38), "Idolaters 

 as well as sun-worshippers existed in ancient Persia, and the 

 wor-hip of fire and that of idols were combined at one period." 



Quintus Curtius, when describing the march of the army of 

 Darius (writing, however, long after date), says : — " Darius was 

 accompanied by an image of the sun, placed in a crystal, and 

 the sacred /f/-f carried on a silver altar." 



The sun, which was regarded as a wheel, a store of gold, an 

 eagle, was also styled the eye of Varuna.' The worship of 

 Mithra was likewise a worship of the sun ; Mithra was the god 

 of daylight. He and Varuna were fabled to journey at even 

 in a brazen car. From this has probably arisen the horse-sun 

 and the wheel-sun. Euripides gives the sun a winged car ; and, 

 on coins from Eleusis, Demeter is represented riding in such a 

 car drawn by two serpents. 



has been very generally allowed 



to be a symbol of Thor, who, to the Scandinavians, was the 

 god of thunder and lightning and of the domestic hearth, and 

 therefore of fire also. The arrows in the hand of Jove, the 

 thunderer of Roman mythology, resemble somewhat a com- 

 pressed or crushed svastika. The above form of this symbol. 



with a very slight variation F^T 



OJT 



be seen on a 



slab taken out of a Christian citacomb in Naples, and now in 

 the National Museum there. A very natural inference is, that 

 this stone sealed the grave of one who had suffered martyrdom 

 by fire. 



The svastika has been held to be an emblem of fire, as being 

 the way in which that element was first produced by primitive 

 peoples— a method which is said to be in use in certain Hindu 

 temples at the present day. It consists in two crooked sticks 

 being laid one across the other, and a hole drilled through both ; 

 a pointed stick being there inserted, this is rapidly twirled by 

 the hands until the five points of contact become ignited. 



' To the Persians, Varuna was the god of the clouds .ind of the celestial 

 sea. When this branch of the Aryans reached Southern India, he there 

 became to them the god of the earthly sea. I'o the Greeks he was Ouranos ; 

 and to the Germans and Anglo-Saxons, the eye of Woden. 



