Franklin Institute-Journal, No. 808. 8vo. 1893. 8vo. 1893. 1893. Harvard University-Bibliographical Contributions, No. 47. 8vo. 1893. Johns Hopkins University-American Chemical Journal, Vol. XV. No. 4. 8vo. 1893. University Circulars, No. 104. 4to. 1893. Keeler, James E. Esq. (the Author)-Observations on the Spectrum of 8 Lyra. 8vo. 1893. Linnean Society-Journal, No. 154. 8vo. 1893. Manchester Geological Society-Transactions, Vol. XXII. Parts 6, 7. 8vo. 1893. Mendenhall, T. C. Esq. (the Author)-Determinations of Gravity. 8vo. 1892. Ministry of Public Works, Rome-Giornale del Genio Civile, 1893, Fasc. 1, 2, and Munir Bey, His Excellency-Catalogue of the Library of the late Ahmed Véfyk Pacha. Constantinople. 8vo. 1893. National Life-Boat Institution, Royal-Annual Report, 1893. 8vo. North of England Institute of Mining and Mechanical Engineers-Transactions, Vol. XLII. Part 2. 8vo. 1893. Odontological Society-Transactions, Vol. XXV. Nos. 5, 6. 8vo. 1893. Payne, W. W. and Hale, G. E. (the Editors)—Astronomy and Astro-Physics for April, 1893. 8vo. Pharmaceutical Society of Great Britain-Journal for April, 1892. 8vo. Richardson, B. W. M.D. F.R.S. M.R.I. (the Author)—The Asclepiad, 1893, No. 1. 8vo. Royal Society of London-Proceedings, No. 320. 8vo. 1893. Smithsonian Institution-Bureau of Ethnology: Contributions to North American Ethnology, Vol. VII. 4to. 1890. Annual Report of the Bureau of Ethnology, 1885-86. 4to. 1891. Society of Architects-Proceedings, Vol. V. Nos. 9, 10. 8vo. 1893. Society of Arts-Journal for April, 1893. 8vo. St. Petersbourg, Académie Impériale des Sciences-Mémoires, Tome XL. No. 2; Tome XLI. No. 1. 4to. 1892-93. Bulletin, Tome XXXV. No. 3. 8vo. 1893. Teyler Museum-Archives, Série II. Vol. IV. Part 1. 8vo. 1893. United Service Institution, Royal-Journal, No. 182. 8vo. 1893. United States Department of the Interior-Report on Mineral Industries in the U.S. at the Eleventh Census, 1890. 4to. 1892. Vereins zur Beförderung des Gewerbfleisses in Preussen-Verhandlungen, 1893, Heft 3, 4. 4to. Victoria Institute-Transactions, No. 103. 8vo. 1893. Zoological Society of London-Proceedings, 1892, Part 4. 8vo. 1893. Transactions, Vol. XIII. Part 5. 4to. 1893. Zurich Naturforschenden Gesellschaft-Vierteljahrschrift, Jahrgang XXXVII. Heft 3, 4. 8vo. 1892. WEEKLY EVENING MEETING, Friday, May 12, 1893. SIR DOUGLAS GALTON, K.C.B. D.C.L. LL.D. F.R.S. The Right Hon. LORD KELVIN, D.C.L. LL.D. Pres. R.S. M.R.I. Isoperimetrical Problems. Dido, B.C. 800 or 900. Horatius Cocles, B.C. 508. Pappus, Book V., A.D. 390. John Bernoulli, A.d. 1700. Euler, A.D. 1744. Maupertuis (Least Action), b. 1698, d. 1759. Lagrange (Calculus of Variations), 1759. Hamilton (Actional Equations of Dynamics), 1834. THE first isoperimetrical problem known in history was practically solved by Dido, a clever Phoenician princess, who left her Tyrian home and emigrated to North Africa, with all her property and a large retinue, because her brother Pygmalion murdered her rich uncle and husband Acerbas, and plotted to defraud her of the money which he left. On landing in a bay about the middle of the north coast of Africa she obtained a grant from Hiarbas, the native chief of the district, of as much land as she could enclose with an ox-hide. She cut the ox-hide into an exceedingly long strip, and succeeded in enclosing between it and the sea a very valuable territory* on which she built Carthage. The next isoperimetrical problem on record was three or four hundred years later, when Horatius Cocles, after saving his country by defending the bridge until it was destroyed by the Romans behind him, saved his own life and got back into Rome by swimming the Tiber under the broken bridge, and was rewarded by his grateful countrymen with a grant of as much land as he could plough round in a day. In Dido's problem the greatest value of land was to be enclosed by a line of given length. If the land is all of equal value the general solution of the problem shows that her line of ox-hide should * Called Byrsa, from Búpoa, the hide of a bull. [Smith's Dictionary of Greek and Roman Biography and Mythology,' article "Dido."] be laid down in a circle. It shows also that if the sea is to be part of the boundary, starting, let us say, southward from any given point A of the coast, the inland bounding line must at its far end cut the coast line perpendicularly. Here, then, to complete our solution, we have a very curious and interesting, but not at all easy, geometrical question to answer :-What must be the radius of a circular arc A DC, of given length, and in what direction must it leave the point A, in order that it may cut a given curve A B C perpendicularly at some unknown point C? I don't believe Dido could have passed an examination on the subject, but no doubt she gave a very good practical solution, and better than she would have found if she had just mathematics enough to make her fancy the boundary ought to be a circle. No doubt she gave it different curvature in different parts to bring in as much as possible of the more valuable parts of the land offered to her, even though difference of curvature in different parts would cause the total area enclosed to be less than it would be with a circular boundary of the same length. The Roman reward to Horatius Cocles brings in quite a new idea, now well known in the general subject of isoperimetrics: the greater or less speed attainable according to the nature of the country through which the line travelled over passes. If it had been equally easy to plough the furrow in all parts of the area offered for enclosure, and if the value of the land per acre was equal throughout, Cocles would certainly have ploughed as nearly in a circle as he could, and would only have deviated from a single circular path if he found that he had misjudged its proper curvature. Thus, he might find that he had begun on too large a circle, and, in order to get back to the starting point and complete the enclosure before night-fall, he must deviate from it on the concave side; or he would deviate from it on the other side if he found that he had begun on too small a circle, and that he had still time to spare for a wider sweep. But, in reality, he must also have considered the character of the ground he had to plough through, which cannot but have been very unequal in different parts, and he would naturally vary the curvature of his path to avoid places where his ploughing must be very slow, and to choose those where it would be most rapid. He must also have had, as Dido had, to consider the different value of the land in different parts, and thus he had a very complex problem to practically solve. He had to be guided both by the value of the land to be enclosed and the speed at which he could plough according to the path chosen; and he had a very brain-trying task to judge what line he must follow to get the largest value of land enclosed before night. These two very ancient stories, whether severe critics will call them mythical or allow them to be historic, are nevertheless full of scientific interest. Each of them expresses a perfectly definite case of the great isoperimetrical problem to which the whole of dynamics is reduced by the modern mathematical methods of Euler, Lagrange, Hamilton and Liouville (Liouville's Journal, 1840-1850). Dido's and Horatius Cocles' problems, we find perfect illustrations of all the fundamental principles and details of the generalised treatment of dynamics which we have learned from these great mathematicians of the eighteenth and nineteenth centuries. In Nine hundred years after the time of Horatius Cocles we find, in the fifth Book of the collected Mathematical and Physical Papers of Pappus of Alexandria, still another idea belonging to isoperimetrics -the economy of valuable material used for building a wall; which, however, is virtually the same as the time per yard of furrow in Cocles' ploughing. In this new case the economist is not a clever princess, nor a patriot soldier; but a humble bee who is praised in the introduction to the book not only for his admirable obedience to the Authorities of his Republic, for the neat and tidy manner in which he collects honey, and for his prudent thoughtfulness in arranging for its storage and preservation for future use, but also for his knowledge of the geometrical truth that a hexagon can enclose more honey than a square or a triangle with equal quantities of building material in the walls," and for his choosing on this account the hexagonal form for his cells. Pappus, concluding his introduction with the remark that bees only know as much of geometry as is practically useful to them, proceeds to apply what he calls his own superior human intelligence to investigation of useless knowledge, VOL. XIV. (No. 87.) 66 I and gives results in his Book V., which consists of fifty-five theorems and fifty-seven propositions on the areas of various plane figures having equal circumferences. In this Book, written originally in Greek, we find (Theorem IX. Proposition X.) the expression "isoperimetrical figures," which is, so far as I know, the first use of the adjective" isoperimetrical" in geometry; and we may, I believe, justly regard Pappus as the originator, for mathematics, of isoperimetrical problems, the designation technically given in the nineteenth century to that large province of mathematical and engineering science in which different figures having equal circumferences, or different paths between two given points, or between some two points on two given curves, or on one given curve, are compared in connection with definite questions of greatest efficiency and smallest cost. In the modern engineering of railways an isoperimetrical problem of continual recurrence is the laying out of a line between two towns along which a railway may be made at the smallest prime cost. If this were to be done irrespectively of all other considerations, the requisite datum for its solution would be simply the cost per yard of making the railway in any part of the country between the two towns. Practically the solution would be found in the engineers' drawing office by laying down two or three trial lines to begin with, and calculating the cost of each, and choosing the one of which the cost is least. In practice various other considerations than very slight differences in the cost of construction will decide the ultimate choice of the exact line to be taken, but if the problem were put before a capable engineer to find very exactly the line of minimum total cost, with an absolutely definite statement of the cost per yard in every part of the country, he or his draughtsmen would know perfectly how to find the solution. Having found something near the true line by a few rough trials they would try small deviations from the rough approximation, and calculate differences of cost for different lines differing very little from one another. From their drawings and calculations they would judge by eye which way they must deviate from the best line already found to find one still better. At last they would find two lines for which their calculation shows no difference of cost. Either of these might be chosen; or, according to judgment, a line midway between them, or somewhere between them, or even not between them but near to one of them, might be chosen, as the best approximation to the exact solution of the mathematical problem which they care to take the labour of trying for. But it is clear that if the price per yard of the line were accurately given (however determined or assumed) there would be an absolutely definite solution of the problem, and we can easily understand that the skill available in a good engineer's drawing-office would suffice to find the solution with any degree of accuracy that might be prescribed; * Example, Woodhouse's Isoperimetrical Problems,' Cambridge, 1810. |