Heron has been said to have a special fondness. In our second process, also, the writer's first auxiliary line seems to have reduced his figure to an obtuse-angled triangle and an isosceles trapezoid.

...

To this view, however, there are one or two serious objections. Perhaps no great importance attaches to the fact that we cannot in Heron find such phrases as ὡς δεῖ, εἰς τὸ αὐτό, for which latter Heron's equivalent is uniformly óμoù; and to the entire absence from Heron's measures of äpovpa. More significant must be considered the inconsistency between Heron's παραλληλόγραμμον and that of the fragment. Heron's parallelogram does not differ from Euclid's: ἐπὶ δὲ τῶν τετραπλεύρων ἃ μὲν καλεῖται παραλληλόγραμμα, . . . παραλληλόγραμμα μὲν οὖν τὰ τὰς ἀπεναντίον πλευρὰς παραλλήλους ἔχοντα, etc., p. 20, ll. 11-13. There appears to be no way of bringing the 'parallelogram' of column III of our fragment within the scope of this definition; and we are confronted with an inconsistency as remarkable as was the agreement in the case of κορυφή. Of Heron's method of calculating the area of a triangle in terms of its sides, our processes show hardly a trace. It has been suggested that our fragment reflects the methods of the second book of Euclid rather than the formula of Heron; but the difficulty of relating our surveyor with Euclidean terminology and method in general, has already been shown to be considerable. The Heronian formula, on the other hand, he clearly had not. Perhaps it is not much to say that he had the materials out of which that formula was derived; but I have thought it not impossible that we have in this fragment one of those early mathematical works of whose materials Heron later became the organizer and compiler; in other words, the work of which this papyrus was a copy, if not itself one of Heron's sources, may fairly represent the character of the sources he had and used.

1

Slight resemblances to other mathematicians may be noted. The fragment's use of inоуeуpaμμévov is somewhat closely paralleled in Apollodorus, ἔστι δὲ τὰ ὑπογεγραμμένα σχήματα, and in Bito, τὸ δὲ σχῆμα οἷόν ἐστιν ὑπογέγραπται; also in a passage ascribed to Heron Byzantinus, καὶ τὸ σχῆμα ὑπογέγραπται. As to ὡς δεῖ, M. Tannery has suggested that it may be a scribal error for us deí, in uncials and a being easily confused. 'Os dei, however, does not seem difficult when compared with Euclid's ὡς ἔτυχεν, ὃ ἔτυχεν, ἃ ἔτυχεν,

3

1 Before fig. 47, Wescher's Poliorcetique des Grecs. 2 Before figs. 17, 19, 20, ibid.

Before fig. 103.

with which expressions it would stand in almost direct contrast, meaning 'by construction,' or perhaps better, 'by the conditions of the problem.'

To facilitate reference to the fragment, the following index has been prepared. Occurrences of kai, dé and the forms of the article are not noticed in this index; nor is the material afforded from the fragments of col. I and of the last process of col. II.

ἄλλο ΙΙΙ 14.

ȧμλvуávov II 14 (see p. 27).

ἄνω III 9.

ȧnó II 5, 6, 7, 8, III 3, 4, 5, 8 bis, 18.

äpovpat, ȧpovpŵv II 1, 2, 10, 11, 13, 14, III 11, 12, 14, 15, 20 bis. avrá W. ép' II 4 bis, 7, III 2, 3, 6 bis, 17 bis..

aurą W. év II 1, 11, 13, III 11, 13.

Els Tò AUTÓ II 1, 14, III 15, 20.

appede II 5, III 3, 6, 8.

ἄφελε

βάσις ΙΙΙ 5.

Báσews II 6, 7, 9, 12, III 3, 8, 11, 13, 17, 19.

ἄνω βάσεως ΙΙΙ 9.

yiverai, yivovraι II 4, 5, 7 ter, 9 bis, 11, 12, 13, 14, III 2, 3, 4, 5, 6

éπí II 9, 10, 12, III 10, 12, 13, 19.

ép' II 4 bis, 7, III 2, 3, 6 bis, 17 bis.

Taι II 2, 14, III 15, 21..

ἑτερόμηκες ΙΙ Ι, 12, ΙΙΙ 13.

μov II 7, 9, 13, III 5, 11, 14, 17, 19.

KáðεTOS II 8, III 7, 18.

καθέτου ΙΙΙ 10.

κορυφής ΙΙ 5, 11, ΙΙΙ 4, 9.

λαβέ ΙΙ 6, ΙΙΙ 4.

λοιπόν, λοιπά, ΙΙ 5, 6, 7, 8, III 3, 4, 5, 7, 8, 9, 18.

λοιπή ΙΙΙ 9.

oin II 3, III 1, 16.

ὀρθογώνιον ΙΙΙ 14.

ὀρογώνιον III 12 (see p. 27); ὀρθογωνίου ΙΙΙ 5, 10; ὀρθογωνίων ΙΙ

III. SEMASIOLOGICAL POSSIBILITIES.

THESIS: Difference in meaning is of itself no bar to connecting words.

If the principle underlying the semasiological development of any word is to be brought to light, it must be by coming at the original meaning of the root and discovering, if possible, the figure of speech involved in its growth. Where the literal meaning is preserved, the task is comparatively easy. If, then, we find an abstract word, we naturally look about for some corresponding concrete term, in the same language or related languages. Quite possibly, in our search, we hit upon two or more concrete words which seem not at all related to each other in meaning. But if they are phonetic equivalents, we should try to trace them to a common origin. For in most cases phonetic equivalents are identical, and difference in meaning no ground for separating them.

Suppose, to illustrate this point, a community shut off from all others has in its vocabulary but two hundred words. What will be the result? Each of these words will be used to express many ideas. One of these words, we will say, is 'strike.' What an endless variety of meanings may arise from this! Even in our copious language we have for this one verb thirty-four distinct uses, as defined by Webster. Each of these significations is capable of further development, as may be easily shown. To give just one or two of these. If 'strike' in our supposed community stands for all kinds of striking, gentle as well as violent, 'strike' could mean 'stroke, caress, treat kindly, love'; or 'strike, stroke, smear, defile'; or 'strike, beat, maltreat, hate'; or 'strike, afflict, distress.' These are only a few of the almost numberless meanings which may spring from this word.

This is exactly what has taken place in the I.E. tongues. I do not mean, of course, to limit the primitive vocables to two hundred, but they were certainly few. Each of these, therefore, must have developed in this same extensive manner, so that the same idea came to be expressed in various ways. In the 'ursprache' there was only a beginning of this growth: in the

separate dialects it continued with ever-increasing luxuriance. Consequently, in different languages, we may find, and should expect to find, words coming from the same root yet differing from each other as widely as 'love' and 'hate,' 'haughty' and 'humble.' If this be the case, and I do not see how any one can withhold assent to this thesis, how, then, shall we proceed when we find words whose roots are phonetically alike, but whose meanings are widely divergent? We should not attempt to derive one meaning from the other, but each from an original.

Etymologists often assume that words change in meaning. They are forced to this conclusion because words which are evidently related have, in different dialects, meanings that are diametrically opposed. Their conclusions are wrong because their premises are not well taken. It must be remembered that words were originally names of concrete things or descriptions of actions. These may be used figuratively, and thus introduce meanings quite distinct from each other, though easily derivable from the original. Or an adjective may be used actively or passively, as Eng. fearful 'terrible' or 'timid.' Aside from this words do not change.

It is not always possible to connect the various meanings that a word may have, since it is often difficult to find out the original force. But the proper method here, as in tracing phonetic changes, is to discover the common source of development. The usual method is to assume that, in a given number of meanings, this one or that is the original. It possibly may be, but the method is entirely wrong. When we compare any set of words, as Skt. pita, Gk. Tariр, Goth. fadar, etc., we do not assume that any one language has preserved the original form without change; nor should we, in comparing meanings, make a like assumption.

Before the fifth edition of his Et. Wtb., Kluge seemed doubtful of the connection between N.H.G. dreist, O.S. thristi and Lat. tristis, although the words are phonetically the same. Now these words contain the element tri-, which is also in Lat. tri-tus 'rubbed, bruised, trodden,' tri-bulum 'threshing sledge.' This tr-i- is an enlargement of the root ter- in Lat. terō and many others. From this root, then, O.S. thristi 'bold' is an active adjective meaning primarily 'oppressing, beating'; while Lat. tristis 'sad' is passive with the original meaning 'oppressed, depressed, down-trodden.' But Lat. tristis is also active in the sense 'stern, harsh, severe,' a natural outgrowth of 'oppressive.'