John Sturgeon Mackay

“ In the case of money, however, the selection of the standard of value, or the unit by which to reckon, constitutes the main, if not the sole, theoretical difficulty to be overcome, previous to the intro duction of a decimal coinage. ”

“ With these excellences are combined a good many defects, some of them inevitable to a system based on a very few axioms and postulates. Thus the arrangement of his propositions seems arbitrary; associated theorems and problems are not grouped together; the classification, in short, is imperfect. That is the main objection to the retention of Euclid as a school-book. ”

“ Proclus preserves a reply made by Euclid to King Ptolemy, who asked whether he could not learn geometry more easily than by studying the Elements " There is no royal road to geometry." Pappus of Alexandria, whose date is rather uncertain, but is probably a century earlier than that of Proclus, says that Euclid was a man of mild and inoffensive temperament, unpretending, and kind to all genuine students of mathematics. ”

“ Practical difficulties would arise from the unwillingness of people to make the changes in thinking and speaking that would be necessitated by new coins, or the altered values of old ones. Of all the schemes proposed in England, that which advocates the retention of the sovereign, or pound sterling, as the unit of value seems to have met with most favour. According to this scheme, the pound would be divided into 10 florins, the florin into 10 cents, and the cent into 10 mils. ”

“ The tenth book is the longest of the Elements. It is occupied with the consideration of commensurable and incommensurable magnitudes, and ends with the proposition that the diagonal and the side of a square are incommensurable. With regard to straight lines, Euclid distinguishes between those which are commensurable or incommensurable in length, and those which are so in power, the latter being defined to be straight lines the squares on which have or have not a common measure. ”

“ The first printed edition of Euclid was a translation of the fifteen books of the Elements from the Arabic, made, it is supposed, by Adelard of Bath (12th century) , with the comments of Campanus of ISTovara. It appeared at Venice in 1482, printed by Erhardus Ratdolt, and dedicated to the doge Giovanni Mocenigo. This edition represents Euclid very inadequately; the comments are often foolish, propositions are sometimes omitted, sometimes joined together, useless cases are interpolated, and now and then Euclid's order changed. ”

“ Here it will be sufficient to state Simson's definition, which is, "A porism is a proposition in which it is proposed to demonstrate that one or more things are given, between which and every one of innumerable other things, not given but assumed according to a given law, a certain relation, described in the proposition, is to be shown to take place ”

“ The twelfth book begins with two theorems of plane geometry, and then discusses chiefly the properties of pyramids, cones, and cylinders. The last two propositions relate to spheres, the last being to prove that spheres have to one another the triplicate ratio of their diameters. In this book is exemplified the method of Exhaustions, which is founded on the principle that by taking away from a magnitude more than its half, from the remainder more than its half, and so on, a remainder is at length reached which is less than any assignable magnitude (book x. prop. 1 ) . ”

“ Later geometers, he says, defined it to be a local theorem wanting part of the hypothesis, but this definition he censures as imperfect. After the publication of Commandine's translation of Pappus (1588) , many attempts were made to extract from this unsatisfactory description a clear idea of what a porism was, and, with the help of the lemmas, to restore Euclid's books. The mystery, which baffled the penetration even of Edmund Halley, was not resolved till the time of Simson, who, in 1722, gained some insight into the subject, and whose posthumous treatise De Poriwnatibus appeared in 1776. ”

“ This being all that is known of the life and character of Euclid, it only remains therefore to speak of his works.
Among those which have come down to us the most remarkable is the Elements (Sroi^eta) . They consist of thirteen books; two more are frequently added, but there is reason to believe that they are the work of a later mathematician, Hypsicles of Alexandria. At the outset of the first book occur the definitions or explanations of the meanings of the terms employed ”

“ The seventh, eighth, and ninth books are arithmetical, that is, treat of the properties of numbers. The definitions relating to them occur at the beginning of the seventh book, and some of these show perhaps the tendency of the Greeks, natural enough to a scientific people with a defective numerical notation, to consider quantity from a geometrical point of view. ”

“ The arguments for and against a change are numerous, and to detail them would be to fill a moderate volume. The principal reason for making the change is that calculation would be enormously simplified, for reduction from one denomination of money into another could always be performed at sight ”

“ A second scheme advocates the adoption of the farthing a:s the unit of value, and its coins of account would be the farthing, the cent or doit = 10 farthings, the florin =10 cents or doits, the pound = 10 florins. The coins required for circulation would probably be in copper, the farthing, the halfpenny, the penny ; in silver, the cent or doit = 2|d., the 2-cent piece or groat = 5d., the shilling = 12|d., and the florin = 25d. ; in gold, the half-sovereign = 10s. ”

“ The propositions, which consist of both theorems and problems, deal with rectilineal figures, principally the triangle and the parallelogram, and the book concludes with the celebrated Pythagorean theorem and its converse. ”

“ It has also been proposed that there should be only two coins of account, the higher equivalent to 100 of the lower, such as florins and cents, the cent in this scheme being the mil of the first. Centesimal coinage similar to this exists in several foreign countries, &c. ; but it is probable that, should a change bs made, the practice of other nations will be imitated only where it is found to conduce to national convenience. ”