A History of Mathematics

"The preface of the second book [of Conic Sections] is interesting as showing the mode in which Greek books were 'published' at this time. It reads thus: "I have sent my son Apollonius to bring you (Eudemus) the second book of my Conics. Read it carefully and communicate it to such others as are worthy of it."

"The first book [of Conic Sections], says Apollonius in his preface to it, "contains the mode of producing the three sections and the conjugate hyperbolas and their principal characteristics, more fully and generally worked out than in the writings of other authors." We remember that Menæchmus, and all his successors down to Apollonius, considered only sections of right cones by a plane perpendicular to their sides, and that the three sections were obtained each from a different cone. Apollonius introduced an important generalisation. He produced all the sections from one and the same cone, whether right or scalene, and by sections which may or may not be perpendicular to its sides. The old names for the three curves were now no longer applicable. Instead of calling the three curves, sections of the 'acute angled,' 'right angled,' and 'obtuse angled' cone, he called them ellipse, parabola, and hyperbola, respectively. To be sure, we find the words 'parabola' and 'ellipse' in the works of Archimedes, but they are probably only interpolations. The word 'ellipse' was applied because y2 < px, p being the parameter; the word 'parabola' was introduced because y2 = px, and the term 'hyperbola' because y2 > px."

"The first book of the Conic Sections of Apollonius is almost wholly devoted to the generation of the three principal conic sections. The second book treats mainly of asymptotes, axes, and diameters. The third book treats of the equality or proportionality of triangles, rectangles, or squares, of which the component parts are determined by portions of transversals, chords, asymptotes, or tangents, which are frequently subject to a great number of conditions. It also touches the subject of foci of the ellipse and hyperbola. In the fourth book, Apollonius discusses the harmonic division of straight lines. He also examines a system of two conics, and shows that they cannot cut each other in more than four points. He investigates the various possible relative positions of two conics, as, for instance, when they have one or two points of contact with each other. The fifth book reveals better than any other the giant intellect of its author. Difficult questions of maxima and minima, of which few examples are found in earlier works, are here treated most exhaustively. The subject investigated is, to find the longest and shortest lines that can be drawn from a given point to a conic. Here are also found the germs of the subject of evolutes and centres of osculation. The sixth book is on the similarity of conies. The seventh book is on conjugate diameters. The eighth book, as restored by Halley, continues the subject of conjugate diameters."

"It is worthy of notice that Apollonius nowhere introduces the notion of directrix for a conic, and that, though he incidentally discovered the focus of an ellipse and hyperbola, he did not discover the focus of a parabola. Conspicuous in his geometry is also the absence of technical terms and symbols, which renders the proofs long and cumbrous."

"The discoveries of Archimedes and Apollonius, says M. Chasles, marked the most brilliant epoch of ancient geometry. Two questions which have occupied geometers of all periods may be regarded as having originated with them. The first of these is the quadrature of curvilinear figures, which gave birth to the infinitesimal calculus. The second is the theory of conic sections, which was the prelude to the theory of geometrical curves of all degrees, and to that portion of geometry which considers only the forms and situations of figures, and uses only the intersection of lines and surfaces and the ratios of rectilineal distances. These two great divisions of geometry may be designated by the names of Geometry of Measurements and Geometry of Forms and Situations, or, Geometry of Archimedes and of Apollonius."

"Besides the Conic Sections, Pappus ascribes to Apollonius the following works: On Contacts, Plane Loci, Inclinations, Section of an Area, Determinate Section, and gives lemmas from which attempts have been made to restore the lost originals. Two books on De Sectione Rationis have been found in the Arabic. The book on Contacts as restored by Vieta, contains the so-called "Apollonian Problem:" Given three circles, to find a fourth which shall touch the three."

"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent."

"Among the earliest successors of Apollonius was Nicomedes. Nothing definite is known of him, except that he invented the conchoid (mussel-like). He devised a little machine by which the curve could be easily described. With aid of the conchoid he duplicated the cube. The curve can also be used for trisecting angles in a way much resembling that in the eighth lemma of Archimedes. Proclus ascribes this mode of trisection to Nicomedes, but Pappus, on the other hand, claims it as his own. The conchoid was used by Newton in constructing curves of the third degree."

"About the time of Nicomedes, flourished also Diocles, the inventor of the cissoid (ivy-like). This curve he used for finding two mean proportionals between two given straight lines."

"Perseus... lived some time between 200 and 100 B.C. From Heron and Geminus we learn that he wrote a work on the spire, a sort of anchor-ring surface described by Heron as being produced by the revolution of a circle around one of its chords as an axis. The sections of this surface yield peculiar curves called spiral sections, which, according to Geminus, were thought out by Perseus. These curves appear to be the same as the Hippopede of Eudoxus."

"Probably somewhat later than Perseus lived Zenodorus. He wrote an interesting treatise on a new subject; namely, isoperimetrical figures. Fourteen propositions are preserved by Pappus and Theon. Here are a few of them: Of isoperimetrical regular polygons, the one having the largest number of angles has the greatest area; the circle has a greater area than any regular polygon of equal periphery; of all isoperimetrical polygons of n sides, the regular is the greatest; of all solids having surfaces equal in area, the sphere has the greatest volume."

"Hypsicles (between 200 and 100 B.C.) was supposed to be the author of both the fourteenth and fifteenth books of Euclid, but recent critics are of opinion that the fifteenth book was written by an author who lived several centuries after Christ. The fourteenth book contains seven elegant theorems on regular solids. A treatise of Hypsicles on Risings is of interest because it is the first Greek work giving the division of the circumference into 360 degrees after the fashion of the Babylonians."

"Hipparchus of Nicaea in Bithynia was the greatest astronomer of antiquity. He established inductively the famous theory of epicycles and eccentrics. As might be expected, he was interested in mathematics, not per se, but only as an aid to astronomical inquiry. No mathematical writings of his are extant, but Theon of Alexandria informs us that Hipparchus originated the science of trigonometry, and that he calculated a "table of chords" in twelve books. Such calculations must have required a ready knowledge of arithmetical and algebraical operations."

"About 155 B.C. flourished Heron the Elder of Alexandria. He was the pupil of Ctesibius, who was celebrated for his ingenious mechanical inventions, such as the hydraulic organ, the water clock, and catapult. It is believed by some that Heron was a son of Ctesibius. He exhibited talent of the same order as did his master by the invention of the eolipile and a curious mechanism known as "Heron's fountain." Great uncertainty exists concerning his writings. Most authorities believe him to be the author of an important Treatise on the Dioptra, of which there exist three manuscript copies, quite dissimilar. But M. Marie thinks that the Dioptra is the work of Heron the Younger, who lived in the seventh or eighth century after Christ, and that Geodesy, another book supposed to be by Heron, is only a corrupt and defective copy of the former work. Dioptra contains the important formula for finding the area of a triangle expressed in terms of its sides; its derivation is quite laborious and yet exceedingly ingenious. "It seems to me difficult to believe," says Chasles, "that so beautiful a theorem should be found in a work so ancient as that of Heron the Elder, without that some Greek geometer should have thought to cite it." Marie lays great stress on this silence of the ancient writers, and argues from it that the true author must be Heron the Younger or some writer much more recent than Heron the Elder. But no reliable evidence has been found that there actually existed a second mathematician by the name of Heron."

""Dioptra," says Venturi, were instruments which had great resemblance to our modern theodolites. The book Dioptra is a treatise on geodesy containing solutions, with aid of these instruments, of a large number of questions in geometry, such as to find the distance between two points, of which one only is accessible, or between two points, which are visible but both inaccessible; from a given point to draw a perpendicular to a line which cannot be approached; to find the difference of level between two points; to measure the area of a field without entering it."

"Heron was a practical surveyor. This may account for the fact that his writings bear so little resemblance to those of the Greek authors, who considered it degrading the science to apply geometry to surveying. The character of his geometry is not Grecian but decidedly Egyptian. ...There are ...points of resemblance between Heron's writings and the ancient Ahmes papyrus. Thus Ahmes used unit-fractions exclusively; Heron uses them oftener than other fractions. Like Ahmes and the priests at Edfu, Heron divides complicated figures into simpler ones by drawing auxiliary lines; like them, he shows, throughout, a special fondness for the isosceles trapezoid. The writings of Heron satisfied a practical want, and for that reason were borrowed extensively by other peoples. We find traces of them in Rome, in the Occident during the Middle Ages, and even in India."

"Geminus of Rhodes (about 70 B.C.) published an astronomical work still extant. He wrote also a book, now lost, on the Arrangement of Mathematics, which contained many valuable notices of the early history of Greek mathematics. Proclus and Eutocius quote it frequently."

"Dionysodorus of Amisus in Pontus applied the intersection of a parabola and hyperbola to the solution of a problem which Archimedes, in his Sphere and Cylinder, had left incomplete. The problem is "to cut a sphere so that its segments shall be in a given ratio.""

"The close of the dynasty of the Lagides which ruled Egypt from the time of Ptolemy Soter, the builder of Alexandria, for 300 years; the absorption of Egypt into the Roman Empire; the closer commercial relations between peoples of the East and of the West; the gradual decline of paganism and spread of Christianity,—these events were of far-reaching influence on the progress of the sciences, which then had their home in Alexandria. Alexandria became a commercial and intellectual emporium. Traders of all nations met in her busy streets, and in her magnificent Library, museums, lecture halls, scholars from the East mingled with those of the West; Greeks began to study older literatures and to compare them with their own. In consequence of this interchange of ideas the Greek philosophy became fused with Oriental philosophy. Neo-Pythagoreanism and Neo-Platonism were the names of the modified systems. These stood, for a time, in opposition to Christianity. The study of Platonism and Pythagorean mysticism led to the revival of the theory of numbers. Perhaps the dispersion of the Jews and their introduction to Greek learning helped in bringing about this revival. The theory of numbers became a favourite study. This new line of mathematical inquiry ushered in what we may call a new school. There is no doubt that even now geometry continued to be one of the most important studies in the Alexandrian course. This Second Alexandrian School may be said to begin with the Christian era. It was made famous by the names of Claudius Ptolemæus, Diophantus, Pappus, Theon of Smyrna, Theon of Alexandria, Iamblichus, Porphyrius, and others."

"Serenus of Antissa was connected more or less with this [Second Alexandrian] school. He wrote on sections of the cone and cylinder, in two books, one of which treated only of the triangular section of the cone through the apex. He solved the problem, "given a cone (cylinder), to find a cylinder (cone), so that the section of both by the same plane gives similar ellipses." Of particular interest is the following theorem [show figure], which is the foundation of the modern theory of harmonics: If from D we draw DF, cutting the triangle ABC, and choose H on it, so that DE:DF=EH:HF, and if we draw the line AH, then every transversal through D, such as DG will be divided by AH, so that DK:DG=KJ:JO."

"In letters which went between me and that most excellent geometer. G.G. Leibniz, ten years ago, when I signified that I was in the knowledge of a method of determining maxima and minima, of drawing tangents, and the like, and when I concealed it in transposed letters involving this sentence (Data æquatione, etc., above cited) that most distinguished man wrote back that he had also fallen upon a method of the same kind, and communicated his method, which hardly differed from mine, except in his forms of words and symbols."

"[Joseph Fourier] carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series \sum_{n=0}^{n=\infty} (a_n\sin nx+b_n\cos nx) represents the function \phi(x) for every value of x if the coefficients a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}\phi(x) \sin nx\,dx, and b_n be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function."