John Sturgeon Mackay

“ The greatest objections to the change, apart from the difficulty of getting people to make it, which is doubtless much exaggerated, are that a decimal system does not admit to a sufficient extent of binary subdivision, and that it does not admit of ternary subdivision at all. The third part, for instance, of a pound, of a florin, of a cent, being 333^, 33J, 3J mils respectively, could not be exactly paid in decimal currency, while there is no difficulty in paying the third part of a pound, or of a shilling by our present coinage. ”

“ The object of the Data is to show that when certain things—lines, angles, spaces, ratios, &c.—are given by hypothesis, certain other things are given, that is, are determinable. The book, as we are expressly told, and as we may gather from its contents, was intended for the investigation of problems; and it has been conjectured that Euclid must have extended the method of the Data to the investigation of theorems. ”

“ 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 question has often been mooted, to what extent Euclid, in his Elements, is a discoverer or a compiler. To this question no entirely satisfactory answer can be given, for scarcely any of the writings of earlier geometers have come down to our times. We are mainly dependent on Pappus and Proclus for the scanty notices we have of Euclid’s predecessors, and of the problems which engaged their attention; for the solution of problems, and not the discovery of theorems, would seem to have been their principal object. ”

“ 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. ”

“ Thus the arrangement of the propositions seems arbitrary; associated theorems and problems are not grouped together; the classification, in short, is imperfect. Other objections, not to mention minor blemishes, are the prolixity of the style, arising partly from a defective nomenclature, the treatment of parallels depending on an axiom which is not axiomatic, and the sparing use of superposition as a method of proof.
Of the thirty-three ancient books subservient to geometrical analysis, Pappus enumerates first the Data (Δεδομένα) of Euclid. ”

“ 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. ”

“ From these authors we learn that the property of the right-angled triangle had been found out, the principles of geometrical analysis laid down, the restriction of constructions in plane geometry to the straight line and the circle agreed upon, the doctrine of proportion, for both commensurables and incommensurables, as well as loci, plane and solid, and some of the properties of the conic sections investigated, the five regular solids (often called the Platonic bodies) and the relation between the volume of a cone or pyramid and that of its circumscribed cylinder or prism discovered. ”

“ 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. ”

“ 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 ) . ”

“ 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 Latin translation, which accompanies the Greek on the same page, is for the most part that of Commandino. The French edition has the title, Les Œuvres d’Euclide, traduites en Latin et en Français, d’après un manuscrit très-ancien qui était resté inconnu jusqu’à nos jours. Par F. Peyrard, Traducteur des œuvres d’Archimède. It was published at Paris in three volumes, the first of which appeared in 1814, the second in 1816 and the third in 1818. It contains the Elements and the Data, which are, says the editor, certainly the only works which remain to us of this ever-celebrated geometer. ”

“ 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. ”

“ Of translations of the Elements into modern languages the number is very large. The first English translation, published at London in 1570, has the title, The Elements of Geometrie of the most auncient Philosopher Euclide of Megara. ”

“ 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. ”