which joins the centres of two circles, that touch each other, passes through the point of contact; a variety of indispensable properties of chords and tangents; and particularly the relation between the angle at the centre of a circle, and those at its circumference, which stand upon the same chord, a proposition that formed, as we shall hereafter see, the basis of the trigonometry of the Greeks. The fourth treats of polygons circumscribed about, or inscribed in the circle. The sixth gives the theory of the proportions of the sides of rectilineal figures. The eleventh contains propositions in respect to planes, solid angles, and the solids themselves, that may be made, as has been done in the treatise we have under consideration, the basis of spherical trigonometry. In all this, however, there is no trace of trigonometry, properly so called. The propositions we have mentioned are absolutely necessary as its basis, but with the exception of the forty-seventh of book first, are of no value in their direct application to calculation.

Hipparchus, although less celebrated than Archimedes, or Euclid, as a geometer, was, perhaps, the most extraordinary proficient in mathematical science of all antiquity. He was little later than Archimedes in point of time, but was possessed of knowledge far more extended. Like Newton, he pointed out a path, and proposed methods, that it required the assiduous labour of centuries to fill up. He first detected the inequa lities of the lunar motion, and in the calculation by which he proves that they exist, as recorded by the most eminent of his successors, he makes use of several of the cases of plane trigonometry. But in his inquiry into the position of those constellations, whose rising is cotemporaneous in the latitude of Rhodes, he manifests his acquaintance with a method of calculating spherical triangles. As this fact is elicited from an examination of his commentary upon the works of one of his most eminent and immediate predecessors, who evidently did not possess this engine of discovery, we consider ourselves warranted in ascribing to Hipparchus, the introduction of spherical, if not of plane trigonometry. As there appears to have been no improvement of any moment made upon the methods of Hipparchus for upwards of a thousand years, an examination into their nature and principles, will not, we trust, be uninteresting to our readers.

The Greeks were unacquainted with that beautiful and simple arithmetical notation, that seems to have been first used in India, but which we owe directly to the Arabs. They were not, however, without one of no small merit, and which wanted, in the hands of one of their calculators, but a single step, to have VOL. 1.-NO. 1.

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made it, in every respect, except simplicity, equal to that now in use. Instead of our nine characters, that represent numbers, they employed the several letters of their alphabet. The first nine of these, corresponded with our digits; the next eight, with an additional character, stood for the series of decimal numbers, from 10 to 90; the remaining eight letters, with a second arbitrary character, represented the series of centurial numbers, from 100 to 900. To express the first nine thousands, they had again recourse to the first nine letters of the alphabet, with a distinctive mark. To represent the myriad, or ten thousand, they made use of its capital initial letter, or M, and the number of myriads was denoted, by means of the letters 'denoting the inferior denominations of the decimal scale. They were thus enabled to express any number less than 1000 myriads; or, in our notation, one hundred millions. Archimedes, in his Arenarius, proposes to take this as a new unit in a second system of numbers, styled by him of the second order; in this way, the expression of any number, less than that which we write by means of a unit followed by sixteen O's, might have been obtained. We, however, know of no instance in which this plan was reduced to practical application. Indeed, the magnitude of their several units of weight, money, and lineal measure, was such, as even at the present day to have required fewer numbers to express such quantity of them, as might become the object of calculation; while in the state of their commerce, and when a much higher value was attached to the precious metals, the first part of the system was amply sufficient for all their purposes. From what has been said, it will be seen, that the system of notation, of the Greeks, was essentially and strictly decimal. But it was less simple than ours, inasmuch as it employed twenty-eight different characters, and wanted that part of our system by which, in consequence of the use we make of the cypher, the value of any number is determined, from the place it holds in the written expression. Although they thus obtained an ascending scale, according to a decimal system, they stopped short of its application to portions of the unit, or had no system of decimal fractions. Their calculations, in ordinary transactions, were hence rendered difficult; but in their trigonometry, would have become impossible, in their notation, from the multitude of numerical symbols that would have been required, had it not been for the happy introduction of a sexagesimal division in the descending scale. The radius of a circle, corresponding in magnitude to the chord of sixty degrees, was by them divided into, sixty equal parts, which probably suggested the farther subdivision

of these parts, each in sixty others; each of the latter, was again subdivided into sixty more, and so on. The radius then was equal to 216,000 of the second order of these subdivisions, and would give them an exactness in their calculations, more than double what is attainable by taking a radius equal to 100,000 units, or by tables of the natural trigonometric functions, to five places of decimals. But such a degree of exactitude was only arrived at by long and painful steps. All their problems were stated as propositions, which required at least one multiplication and a division in their solution. Multiplication and division, in their sexagesimal method, was performed upon a principle we still apply to the former, in a few cases of complex fractions, and which goes, in our common books of arithmetic, by the name of Cross Multiplication.

To prevent the accumulation of work, the lower denominations of the products were neglected in each successive operation. The basis of the solutions of the cases of plane trigonometry, rested upon the fact, that any triangle can be circum-. scribed by a circle; each of its sides, then, becomes the chord of an arc, and the sum of the three arcs, is the whole circumference; and as each of these arcs, is the measure of the angle at the centre, which is double the angle at the circumference, each side is, of course, the chord of twice the opposite angle. If then the sides were expressed in any known measure, they would be to each other, in the ratio of the chords of twice the opposite angles; and by the analogies that may be deduced from this proposition, every case of plane trigonometry is soluble. To facilitate their calculations, tables were constructed, containing the value of the chords, in terms of the sexagesimal division of the radius.

From this fundamental proposition of plane trigonometry, every case of spherical might have been deduced and solved, by means of the table of chords; but three of these cases appear to have escaped the investigation of the Greeks, namely, those where two sides and the included angle, the three sides, and the three angles, are given. Their most improved modes of calculation were, therefore, not only painful from their length, but incomplete in their application to the cases. In this state, trigonometry appears to have rested, from the time of Hipparchus, until that of the Arabians, a space of eleven hundred years. There has come down to our day, a monument that exhibits the extent to which graphic processes, founded upon these principles, were carried; the temple of the Winds, at Athens, has upon its eight faces, as many dials, and the construction of these is so true, as to show as great a de

gree of accuracy as the case is susceptible of; more, indeed, could not, nor will be done, by the methods of the moderns.

The most extensive collection of the trigonometry of the Greeks, is to be found in the works of Ptolemy, an astronomer of the city of Alexandria. We also find some few details, but of far less importance, in the treatises of Menelaus, and Theodosius; all of these were subsequent to the commencement of our era, and more than two hundred years later than Hipparchus, from whom, it is evident, that they derived all their Imethods of calculation.

The next important improvement in trigonometry, that we have to notice, was made by the Arabs, under the reign of the Caliphs. One of these, Al-Mamoun, the son of Haroun Alraschid, so famous in fictitious history, was a most munificent patron of literature and science. He collected, at great pains and expense, the works of the Greek Mathematicians, and thus laid the foundation of a school of the science, that flourished, in the vast regions over which the Saracen arms extended their sway, for near five centuries. A few years after his death, we meet with the name of Albategni, an Arab prince, who not only made himself master of all the trigonometrical knowledge of the Greeks, but extended it in a most important direction. In Ptolemy's account of the use of an instrument called the Analemma, intended for the purpose of facilitating the construction of dials, we find him using the half chords, instead of the chords themselves. But this is an operation purely graphic and experimental, and there is no trace of their having been used by him for the purpose of calculation. Albategni appears to have derived from this, the hint of his important improvement. The diameter, says he, which divides an arc of a circle into two equal parts, divides its chord into two equal parts also, which have each to the radius, the same ratio that the chord has to the diameter. In calculations, then, instead of doubling the angle of a triangle, and using the chord of the double arc, the simple angle may be used, by means of the half chord. These half chords were introduced by him into his calculations, and tables constructed to render their employ. ment more easy. To these half chords, we apply the name sine; some have supposed, that this term is an abbreviation of the method of writing semis inscriptarum, s. ins; but Delambre, with more probability, states, that the Latin sinus, is a literal translation of the name given to them by the Arabs. Albategni also discovered the properties, and made use of versed sines. His tables were still sexagesimal, being constructed by a binary division of the chords of the Greeks. In addition to

these improvements in the method of calculation, we find him in possession of the theorem, by which the angles of a spherieal triangle may be determined, when the sides are given; a theorem, as has been seen, unknown to the Greeks.

Ebn Jormis, who followed Albategni, at an interval of about a century, was in possession of the mode of using the tangents, and secants, and what is still more remarkable, appears to have employed, in order to simplify his calculations, subsidiary arcs, now so frequently employed in astronomical calculations, but which were not re-invented in Europe, before the middle of the eighteenth century. To these tangents, the Arabs gave the name of shadows, thus referring their origin to the use of the gnomon, the earliest of all astronomical instruments, and in which the properties of the tangent are so obvious, that it appears strange that they could have so long escaped notice. The tangents of Ebn Jormis were calculated to a radius of 12 digits, but his contemporary, Aboul Wefa, made them commensurable with the sines, by introducing the sexagesimal division. It will be thus seen, that the Arabs were acquainted with all the trigonometrical functions that are at present employed, and, indeed, of two, the versed sine and secant, that however valuable in their restricted methods of calculation, are no longer of any real use in practical trigonometry.* Decimal arithmetic, in the form we now employ it, had also reached them from Hindostan, but they did not make use of it in their trigonometrical calculations. This has been made to them a subject of reproach, but in truth, before the introduction of logarithms, the application of the decimal scale is rather an increase of the labour of calculation, than any actual facility.

The first cultivators of trigonometry, in modern Europe, were rather the translators and commentators of the Arabians, than either inventors or improvers. Among them are to be named, as the most eminent, Purbach, Reinhold, Maurolycus and Regiomontanus. But they were not aware of the value and extent of the principles on which they rested their methods. This is most remarkable, in the case of the tangents, the importance of which they do not appear to have understood, and in whose use they appear to have made less progress than their predecessors, the Arabs. It is in the works of Vieta, that we first discover a complete system of trigonometry. He lived towards the close of the sixteenth century, and of all the authors who have written upon this subject, with the single exception

• We do not pretend to derive our acquaintance with the trigonometry of the Arabs from their own works, even in translation, but have to acknowledge our obligations to Delambre's treatise.