HI 



OF.OMF.TRY. 



GEOMETRY. 



of astronomers [AsTBosoHTl, and the supposition that the rue 

 of the Nile obliged the builders of the pyramid* to make new land- 

 mnrki one* a year, requires at taut contemporary eridence to make it 

 history. At the came time, the question of the actual origin of geo- 

 metry is a very difficult one, and any conclusion can only be of very 

 moderate probability. 



Among the Chinete, the Jesuit mUaionarie* found very little know- 

 ledge of the propertiea of ipaoe ; a few rule* for mensuration, ami the 

 famous property of the right-angled triangle being all that they could 

 certain. Of all the book* which Oaubif could find proteasing to be 

 written before B.C. 206, there is only one which contains anything 

 immediately connected with geometry. From this writing (called 

 Tcheou-pey) it is not very certain whether the Chinese possetmd the 

 property of the right-angled triangle generally, or only one particular 

 oase ; namely, when the sides are 3, 4, and 6 : and nothing appears 

 which directly or indirectly resembles demonstration. The Hindoos 

 produce a much larger body of knowledge, but of uncertain date. 

 The works of Brahmegupta and Bhascara, of the 7th and 12th centuries 

 after the Christian ern (according to Colebrooke), contain a system of 

 arithmetical mensuration which is certainly older than the compilers 

 mi ntioned, and in which the property of the right-angled triangle is 

 made to produce a considerable number of results ; for instance, the 

 method of finding the area of a triangle of which the three sides are 

 given. By a figure drawn on the margin of some manuscripts, it 

 appears that a demonstration of the property in question had been 

 obtained. [HTPOTHRHUSE.] The circumference of the circle is given 

 as bearing to the diameter the proportion of 3927 to 1250 by the latter 

 writer; being exactly that of 3.1416 to 1. BrahmeguptA takes the 

 proportion of the square root of 10 to 1, or 8'16 to 1. The superior 

 correctness of the later writer could not have arisen from any inter- 

 mediate communication with Kurope, since the true ratio was not 

 known so near as 3'141t> till after the 12th century; and the Persians 

 (as appears by the work of Mohammed ben Mviwi) had adopted this 

 ratio from the Hindoos, before the discovery of an equally exact ratio 

 in Europe. This subject is noticed more in detail in the Article VIOA 

 OAHITA in the Bioo. Div. ; here we merely observe that though no 

 date can be fixed to the commencement of geometry in India, yet the 

 certainty which we now have that algebra and the decimal arithmetic 

 came from that quarter, ' the recorded visits of the earlier Greek 

 philosophers to Hindustan (though we allow weight rather to the ten- 

 dency to suppose that philosophers visited India, than to the strength 

 of the evidence that they actually did so), together with the very 

 striking proofs of originality which abound in the writings of that 

 country, make it essential to consider the claim of the Hindoos, or of 

 their predecessors, to the invention of geometry. That is, waiving.the 

 question whether they were Hindoos who invented decimal arithmetic 

 and algebra, we advance that the people which first taught those 

 branches of science is very likely to have been the first which taught 

 geometry ; and again, seeing that we certainly obtained the former two 

 either from or at least through India, we think it highly probable that 

 the earliest European geometry also came either from or through the 

 same country. 



Of the Babylonian and of the Egyptian geometry we have no 

 remains whatever, thongh each nation has been often said to have 

 invented the science. In reference to the authorities mentioned above 

 in favour of the Egyptians, to whom we may add Diogenes Laertius, 

 Ac., we may say that no one of the writers who tells the story in 

 question is known as a geometer except Proclus, the latest of them all; 

 and as if to give the assertion the character of an hypothesis, this last 

 writer also odds that the Phenicians, on account of the wants of their 

 commerce, became the inventors of arithmetic. In the Jewish writings 

 there is no trace of any knowledge of geometry. So that allowing the 

 Greeks to have received the merest rudiments either from Egypt or 

 India, or any other country, it is Impossible to name any quarter from 

 which we can with a shadow of probability imagine tlirm t have 

 received a deductive system, to ever so small an extent. That their 

 geometry, or any of it, came Hirrri from India, is a supposition of some 

 difficulty : those who brought it could hardly have failed to bring 

 with it the decimal notation of arithmetic. That Pythagoras 

 travelled into India, is (according to Stanley) only the asset! 

 Apuleius and Clemens Alexandrinus, though rendered fprobable by 

 several of his tenets: the better authorities carry him no farther 

 !': ,'. r..-yi' 



Thale* (800 B.C.) and Pythagoras (640 B.C.) founded the earliest 

 schools of geometry. The latter is said to have sacrificed a hecatomb 

 when he discovered the property of the hypothenuse before alluded to, 

 and this silly story is repeated whenever the early history of geometry 

 is given. A large collection of miscellanies might easily be made from 

 the works of writers who were not themselves acquainted with 

 geometry ; but, rejecting such authorities, we shall content ourselves 

 with citing Pappus and Proclus, both geometers, who, living in the 

 fourth and fifth centuries after Christ, had abundant opportunities of 

 hearing the stories to which we allude, and of receiving or rejecting 

 them. 



According to Proclus (book ii. ch. 4, Comm. in Eucl.) Pythagoras 

 was the first who gave geometry the form of a science, after whom 

 came Anaxagoras, CEnopides, Hippocrates of Chios (who invent. 1 the 

 well known quadrature of the lunules), and Theodora* of Cvrcue 



Plato was the next great advancer of the science, with whom were 

 contemporary Leodama*, Archytu, and Thecetetus, of Thasus, 

 Tarentum, and Athens. After Laodamas came Neoclides, whose 

 disciple Leo made many discoveries, added to the accuracy of the 

 elements, and gave a method of deciding upon the |>omibility or 

 impossibility of a problem. After Leo came Kudoxus, the fri 

 Plato, who generalised various results which came from the sel 

 the latter. Amyclas, another friend of Plato, and the brothers 

 MetUBchmua and Dinostratus made geometry more perfect. Thru. linn 

 wrote excellent elements, and generalised various theorems. Cy'u-inus 

 of Athens cultivated other parts of mathematics, but parti 

 geometry. Hermotimus enlarged the results of Eudoxtis and The., 

 and wrote on lad. Next is mentioned Philippus, and after him Ku. h.l, 

 'who was not much younger than those mentioned, and who put 

 together elements, and arranged many thing* of Eudoxus, and gave 

 unanswerable demonstrations of many things which had been 

 demonstrated before him.' He lived under the first Ptolemy, by whom 

 he was asked for an easy method of learning geometry, to which he 

 made the celebrated answer that there was no royal rood. He was 

 younger than the time of Plato, and older than Eratosthenes and 

 Archimedes. He was of the Platonic sect. 



Such is, very nearly entire, the account which Proclus gives of the 

 rise of geometry in Greece. 



Before the time of Euclid demonstration had been introduced, about 

 the time, perhaps by the instrumentality, of Pythagoras ; pure geo- 

 metry had been restricted to the right line and circle, but by whom is 

 not at all known ; the geometrical analysis, and the study of tl 

 sections, as also the consideration of the problems of the duplication <.f 

 the cube, the finding of two mean proportionals, and the trisection of 

 the angle, had been cultivated by the school of Plato ; the quadrature 

 of a certain circular space had been attained, and the general ]>r< 'I -1. in 

 suggested and attempted by Hippocrates and others; a curve of double 

 curvature had been imagined and used by Archytas ; writings existed 

 both on the elements, and on conic sections, loci, and detached 

 subjects. The little that is known of the biography of EUCLID or 

 ALEXANDRIA, will be found in the Bioo. Div. 



Besides the Elements, Euclid wrote, or is supposed to have written, 

 the following works : 



1. 'ZvyypawiM VevSafluv, a treatise on Fallacies, preparatory to 

 geometrical reasoning. This book, mentioned by Proclus, does not 

 now exist, and there is no Greek work of which we so much regret 

 the loss. Hod it survived, mathematical students would not have 

 been thrown directly upon the Elements, without any pi 

 exercise in reasoning. [MATHEMATICS.] 



2. Four books of Conic Sections, afterwards amplified and appro- 

 priated by Apollonius, who added four others. So says Pappus. That 

 Euclid did not write any books on conic sections that he wrote these 

 cf Apollonius is wholly incredible appears to us more than probable 

 from the silence of Proclus the Platonist, who, eulogising Euclid the 

 Platonist, and stating that he wrote on the regular solids (a part of 

 geometry cultivated by the Platonist*), being led thereto by I'latonism, 

 never mentions his writing on the still more Platonic subj. 



the conic sections. But that Aristtcus had written on the subject 

 is known, and that Euclid taught it cannot be doubted, any more 

 than that Apollonius, like other writers, prefixed to his own dii- 

 coveries all that he judged fit out of what was previously known on the 

 subject. 



3. npl Aioipto-f r, on Divisions. This work is mentioned by Proclus 

 in two words. John Dee imagined the book of Mohammed of Bagdad 

 (which is annexed to the English edition of Kuclid hereinafter cited) 

 on the divisions of surfaces to be that of Euclid now under considera- 

 tion; but there seems to be no ground for this notion. The Latin of 

 this work (from the Arabic) is given at the end of Gregory's Euclid, 

 together with a fragment ' Do Lcvi et Ponderoso,' attributed, without 

 any foundation, to Euclid. 



4. npl *o/>i<tyurra>, on Porisms, in three books. This is mentioned 

 both by Pappus and Proclus, the former of whom gives the enunciations 

 of various propositions in it, but the text is so corrupt that they 

 can hardly be understood. On this singular question, see the article 



1'OIIISM. 



6. Timm p4t iwifJarfiav, Locorum ad Superficiem : which we can- 

 not translate. It is mentioned by Pappus, but has not come down 

 to us. 



The preceding works are either lost or doubtful; those which follow 

 all exist, and are contained in Gregory's edition, in the order inverse 

 to that in which they are here mentioned. 



6. 'OirriKa Kal xaromptxd, on Optics and Catoptrics. These books 

 are attributed to Euclid by Proclus, and by Marinus in the preface to 

 the Data; or rather books on these subject*. Savile, Gregory, and 

 others doubt that the books which have come down to us are those of 

 Kuelid, and Gregory gives his reasons in the preface, which arc that 

 Pappus, though he demonstrates proposition* in optics and also in 

 astronomy, and mentions the Phenomena of Euclid with reference to 

 the latter, does not mention the Optics with reference to the former 

 and that there ore many errors In the works in question, such as it is 

 not likely Euclid would have made. Proceeding on the supposition 

 that rays of light are carried from the eye to the abject, the first of these 

 book* demonstrates some relations of apparent magnitude, and shows 



