crown.

respecting hyperbolic conoids and spheroids. 3. A Tract on Spirals, The curve known by the name of the Spiral of Archimedes was originally discovered by his friend Conon, whose premature death prevented his completing the investigation of the properties of the curve. Archimedes completed the investigations, and put them forth as they appear in this tract. Besides his writings on Geometry, we have a tract on Arithmetic, entitled Yaμμiτns, or Arenarius. The object was to prove the possibility of expressing the number of grains of sand which would fill the whole space of the universe considered as a sphere extending to the stars. In this tract he alludes to a system of numeration which he had discovered, and which he had described in a work addressed to Zeuxippus. Two books are also extant on the Equilibrium of Planes, and on their centres of gravity, in which he has proved the fundamental property of the Lever, and shewn how to find the centre of gravity of a triangle, and other figures. Two books, "On Bodies which are carried in a fluid," in which the general conditions of a body floating on a fluid are investigated and applied to different forms of bodies. Archimedes was also the discoverer of the method of finding the specific gravity of bodies. The story of the crown of king Hiero, to whom Archimedes was related, is briefly this. Hiero had delivered to a goldsmith a certain weight of gold to be converted into a votive The king suspected that the crown he received from the smith was not of pure gold, though of the proper weight, but that it was alloyed with silver. He applied to Archimedes to ascertain, without melting the crown, whether it contained alloy. It was observed by Archimedes, on going into a bath full of water, that when his body was immersed in the bath, a quantity of water equal to the bulk of his body flowed over the edge of the bath. It occurred at once to him, that if a weight of pure gold equal to the weight of the crown were immersed in a vessel full of water, and the quantity of water left in the vessel measured, on the gold being taken out; by doing the same with the crown in the same vessel, he would be able to ascertain whether the bulk of the crown were greater than the bulk of an equal weight of pure gold. For any weight of silver is larger in bulk than an equal weight of pure gold. According to Vitruvius, as soon as he had discovered the method of solution, he leaped out of the bath, and ran hastily through the streets to his own house, shouting evρηкα, eйρnкa! He was also the inventor of a machine for raising water from lower to higher levels, which was called the screw of Archimedes. An important application of the principle of this screw has lately been made in the propulsion of ships by means of steam power. When Syracuse was besieged by a land and naval armament under Appius and Marcellus, the besieged held out a successful resistance for three years chiefly by the aid of the machines invented by Archimedes. The city was at length surprised and taken B. c. 212. Archimedes, when seventy-five years of age, was slain by a soldier, while intent on the solution of a problem. He is reported to have expressed a desire that a sphere inscribed in a cylinder might be engraved on his tomb, to record his discovery of the relation between the volumes and the surfaces respectively of these two solids. About 200 years after his death, his tomb was discovered near Syracuse by Cicero, while quæstor in Sicily. It was nearly overgrown with bushes and brambles, which he caused to be cleared away. The tomb was identified as the tomb of Archimedes by

the Sphere and Cylinder engraved upon the stone with the inscription, the latter part of which was completely effaced. (Cic. Tusc. Quæst. lib. v.)

Conon was the friend and cotemporary of Archimedes, and is celebrated by Virgil in his third Eclogue. In the treatise on the quadrature of the parabola, speaking of his genius, Archimedes exclaims"How many theorems in geometry, which to others have appeared impossible, would Conon have brought to perfection, if he had lived!"

Cotemporary with Archimedes was Eratosthenes, a distinguished geometrician and astronomer. He is celebrated for his construction in solving the problem of the duplication of the cube. He was the first who attempted to measure the circumference of the earth by means of observations of the Sun at two different places, near the same meridian, at the time of the solstice. Though he did not completely succeed, on account of the inaccuracy of his data, he pointed out the method. None of his works have descended to modern times except a few fragments, and a list of the names of forty-four constellations, and the principal stars in each constellation.

Apollonius of Perga, a city of Pamphylia, lived about the same time, and stands next in fame to Archimedes. He was born at the time when Ptolemy Euergetes was king of Egypt: he studied the mathematical sciences at Alexandria in the school which Euclid's disciples had founded, and passed there the greater part of his life. He was the author of several works on Geometry, and became so eminent in that science that he was called by his cotemporaries the Great Geometer. His principal work is a treatise on the Conic Sections. From the author's dedicatory epistle to Eudemus, a geometer of Pergamus, it appears that the treatise consisted of eight books. The first four books still exist in the original Greek. An Arabic version of seven books, made about the middle of the thirteenth century, was found in the East about four centuries later by Golius, a professor of the oriental languages at Leyden; and was translated into Latin. It has been said that Apollonius appropriated the discoveries of others in the Conic Sections. It is well known that although Archimedes discovered many important properties of the curves which bear that name, it cannot be pretended that he was the original discoverer. Long before the time of Archimedes they had been studied in the school of Plato and at Alexandria, and many properties of them were well known. It is highly probable that Apollonius, in his treatise on the Conic Sections, availed himself of the writings of Archimedes as well as of others who had gone before him. His treatise on Conics was most highly esteemed; and to him is justly accorded the honour of having composed a better treatise on that difficult subject than any who had written before him. He made important improvements in the problems both of Euclid and of Archimedes. Before Apollonius, we are informed, by his commentator Eutocius, that writers on the Conic Sections required three different sorts of cones from which to cut the three different sections. They used to cut the parabola from a right-angled cone, the ellipse from an acute-angled cone, and the hyperbola from an obtuse-angled cone: because they always supposed the sections made by the cutting planes to be at right angles to the side of the cone. But Apollonius cut his sections from any cone by only varying the inclination and position of the cutting plane. It may be remarked that Apollonius first gave the names of ellipse and hyperbola to two of the

curves: the name of parabola had been already applied to the third by Archimedes. He also first made the distinction between the diameters of the sections, and the axes, giving the latter name to the two diameters which are at right angles to each other in the ellipse and hyperbola, and restricting the term axis in the parabola, to the line which passes through the focus and vertex of that curve. The following is a very brief account of the subject of each book of this treatise.

Book 1. treats of the generation of the Conic Sections and their distinguishing properties.

Book II. treats of the properties of diameters and axes of these three curves, and of the asymptotes of the hyperbola.

Book III. consists of Theorems useful in the solution of solid loci. Book IV. explains his new method of the intersection of the sections of cones with each other and with the circumferences of circles. Book v. treats of maxima and minima in the Conic Sections. Book VI. treats of equal and similar sections of the cone.

Book VII. contains a collection of Theorems useful in the solution of Problems.

Book VIII. was a collection of Problems with their solutions by means of the Theorems in Book VII.

Besides the treatise on the Conic Sections, Apollonius was the author of several treatises relating to the Geometrical Analysis. They bear the following titles in the translation of Pappus's Collections, and each treatise consisted of two Books.

1. De Rationis Sectione.

2. De Spatii Sectione.

3. De Sectione Determinatâ. 4. De Tactionibus.

5. De Inclinationibus,

6. De Planis Locis.

An Arabic Version of the Treatise De Sectionis Ratione was translated into Latin by Dr Halley and published at Oxford in 1708. The rest are not known to be extant either in the original Greek or in Arabic. Attempts however have been made since the revival of learning in Europe to restore these lost treatises; notices of which will be found under the names of the respective authors in their proper places. Proclus, in his commentary on Euclid, informs us that Apollonius attempted to demonstrate the axioms of Euclid, and cites his method of proving the first axiom, that things which are equal to the same thing are equal to one another. Proclus examines his so-called proof, and shews that properties are assumed not more self-evident than the axiom itself.

Nicomedes lived during the second century before the Christian era, and is known for the invention of a curve called the conchoid, and for the application he made of it in finding two mean proportionals between two given lines. He was also celebrated for the invention of several useful machines.

About the same time also lived Nitocles, Thrasideus, and Dositheus, whose discoveries in mathematics, and whose writings, if they left any, have not descended to our times.

Geminus, a native of Rhodes, and a mathematician of some repute, lived about 100 years before the Christian era, and is reported to have been the author of a work entitled Enarrationes Geometricæ,' which is not known to be extant.

Hipparchus, a native of Nice, though not a writer on the Elements of Geometry, is regarded as the first who reduced Astronomy to a science, and either devised or greatly improved the methods of calculation in Trigonometry, which form the basis of the science of Astronomy. His work on the calculation of chords originally consisted of 12 books, of which a few fragments only are known to be extant. He divided the circumference of the circle into 360 equal parts, and also the radius into 60 equal parts, which he likewise called degrees, each degree into 60 parts, and so on. His rules of calculation were derived from the properties of chords, and were estimated in sexagesimal parts of the radius, and the lengths of chords were calculated to every half degree of the semicircumference. It is, however, as an astronomer that his name is most celebrated. He was the first who discovered the precession of the equinoxes, and taught how to foretell eclipses, and form tables of them. The catalogue of stars which he observed and registered between the years B. c. 160 and 135, is preserved in Ptolemy's Almagest. They are arranged according to their longitudes and their apparent magnitudes. He was also the first who suggested the idea of fixing the position of places on the earth, as he did in the heavens by means of their latitude and longitude. He pursued his astronomical studies at Rhodes, whence he obtained the name of Rhodius, and afterwards in Bithynia, and at Alexandria. His writings on Astronomy are highly spoken of by ancient authors, but are not now extant: his commentary, however, on the Phænomena of Aratus, still exists. A full but rather exaggerated account of the discoveries of Hipparchus will be found in the work of Delambre, on the ancient Astronomy of the Greeks.

a

There is some uncertainty with respect to the exact period when Hypsicles flourished. He was a native of Alexandria, and, it is said, disciple of Isidorus. To him are attributed the 14th and 15th books of the Elements, which were added to the 13 books of Euclid. In the introduction, he makes mention of Apollonius, who flourished in the reign of Ptolemy Euergetes, and probably Hypsicles came after him.

Theodosius of Tripoli flourished about the time of Cicero, and was the author of a treatise on the sphere in three books, which has come down to our time, in the original Greek. In this treatise, he investigates the properties of circles, which are made by sections of the surface of the sphere. It was translated and published by Dr Barrow in 1675: another edition was published at Oxford, in 1707.

For a period of some hundreds of years after the time of Theodosius, we shall find that few additional discoveries were made in geometrical and mathematical science. There were, however, some instances of individuals during that period, not entirely unskilled in Grecian science. Sosigenes the peripatetic, was a mathematician and astronomer. He was an Egyptian, and was brought by Julius Cæsar to Rome, for the purpose of assisting in the reformation of the Roman Calendar. This philosopher had discovered, by astronomical observation, that the year consists of 365 days and 6 hours; and to make allowance for the accumulation of the hours which were above 365 whole days, he invented the intercalation of one day in four years. The duplication every fourth year of the sixth day before the Calends of March was the intercalary day; and hence the year in which this took place consisted of 366 days, and was named bissextile. This was called

the Julian correction of the Calendar, and the reckoning by Julian years commenced B. c. 45, and continued till the more accurate correction was made under Pope Gregory XIII. Marcus Vitruvius was also greatly esteemed by Julius Cæsar, and he was subsequently employed by Augustus in constructing public buildings and warlike machines. Vitruvius was the author of a work on Architecture, in ten books, which was addressed to Augustus. This work is still extant, and is the only one on the Architecture of the ancients, which has descended to modern times. It affords evidence in the ninth book, that the writer was well skilled in Geometry.* Menelaus was born at Alexandria, in the time of the Emperor Trajan, and was of Grecian origin. He composed a Treatise on Trigonometry in six books, and another on the Sphere in three books, both of which have come down to us through the medium of translations in Arabic. A Latin translation of the Spherics was published at Paris in 1664.

Claudius Ptolemæus, one of the most eminent mathematicians and

The Romans were a nation of warriors, and at no period of their history distin. guished for their cultivation or advancement of the sciences. Though it must be admitted that their history exhibits many noble instances of patriotism and the sterner virtues, the leading principle of the Roman policy was nothing less than universal dominion. In the life of Agricola, Tacitus has recorded the substance of the address of Calgacus to the Northern Britons when invaded by the Romans, in which are the following lines: "Sed nulla jam ultra gens, nihil nisi fluctus et saxa, et infestiores Romani: quorum superbiam frustra per obsequium et modestiam effugeris: raptores orbis, postquam cuncta vastantibus defuere terræ, et mare scrutantur: si locuples hostis est, avari: si pauper, ambitiosi: quos non Oriens, non Occidens, satiaverit: soli omnium, opes atque inopiam pari affectu concupiscunt: auferre, trucidare, rapere falsis nominibus, imperium; atque, ubi solitudinem faciunt, pacem appellant." To this may be added the following passage from the popular work of M. Aimé-Martin; "La règne de Rome fut celui d'un brigand; elle s'aggrandit par la guerre et la pillage; et aussi elle perit par ses richesses et par la guerre." This will scarcely be deemed too highly coloured a description of Roman policy and practice, when it is compared with the following lines which Virgil puts into the mouth of Anchises.

"Excudent alii spirantia mollius æra,

Credo equidem; vivos ducent de marmore vultus;
Orabunt causas melius, cœlique meatus

Describent radio, et surgentia sidera dicent:

Tu regere imperio populos, Romane, memento:

Hæ tibi erunt artes; pacisque imponere morem,

Parcere subjectis, et debellare superbos." (En. vI. 847.)

These lines were written when nearly the whole world was subjected to the Roman arms, and when the future prospect of undisputed sway appeared to threaten that the dominion of Rome would be universal and perpetual. It may seem surprising that even in the golden age of Roman literature and magnificence, there does not appear to have existed one Roman of original genius, who successfully cultivated and advanced the mathematical sciences. The slender amount of scientific knowledge attained at that period, was acquired at Alexandria and at Athens, which seem to have been the chief places of resort for the philosophers and their hearers. Horace humorously concludes his description of the course of his education by declaring, that the object for which he was sent to Athens by his father was merely

"Scilicet ut possem curvo dignoscere rectum."

From this, perhaps, we may infer the opinion of that age, that a little mathematical knowledge was not deemed wholly useless, or incompatible with the pursuit of literature. With regard to the knowledge of astronomy which existed among the Romans, it may be observed, that it was rather cultivated for its supposed utility in relation to astrology and the prognostication of future events, than for its real value as science.