in modern Europe until a comparatively late period, and was obstinately retained as long as the increasing light of inductive truth still left a dark nook where its partisans could find refuge in congenial obscurity.

Of the instruments commonly employed by the ancient astronomers, the gnomon was at once the simplest, and probably the most accurate for its purpose. Besides this, we find them, especially in these later ages, using astrolabes, and armillary spheres of various constructions, which being fixed with their circles in the actual positions of the real sphere, the equinoxes were observed by the coincidence of the plane of the circle representing the equator with its own shadow. Altitudes were measured by a circle in the plane of the meridian, and the shadow of a small projecting piece at the extremity of a revolving diameter, made to fall exactly on a corresponding projection at its other extremity. Ptolemy himself contrived a quadrant with a similar apparatus. He also used an instrument of a similar principle, but of which all the parts were rectilinear: he thus measured the chord of the angle observed; and found the angle by a table of chords. The great defect in all these instruments was that want of certainty in fixing upon the precise position of any celestial object, which could not be remedied without a knowledge of the telescope. Their graduation also, probably, was but of a very inferior degree of accuracy. Another fundamental deficiency in their observatories was in the accurate means of measuring time. The clepsydra was open to manifest objection. Upon the whole, it is rather matter of astonishment that, in the department of actual observation, the ancients should have done so much, considering the limited means at their command, than that they should not have effected more.

Optics.

The labours of Ptolemy were not confined to astronomy: he produced a treatise on Optics; the first in

which the subject of refraction is accurately enquired into. This treatise, though known in the middle ages, and quoted by Roger Bacon, had disappeared, and was supposed to be entirely lost, till, in very recent times, a copy (professedly a translation from the Arabic) was found in the king's library at Paris: another copy also exists in the Savilian library at Oxford.

Ptolemy, actuated, doubtless, by the necessity he felt, as an astronomer, for a more accurate knowledge of atmospheric refraction, as affecting the places of the heavenly bodies, examined with great care and precision the angles of refraction corresponding to all angles of incidence from 0 to 80 degrees, when a ray enters a a medium of water, or of glass, out of air. The theory which he hence gave of astronomical refractions was even more correct than that adopted by some of the moderns. In an optical point of view, his measurements accord very exactly with the modern law of the sines: though he did not advance to any such generalisation from them.

Some modern writers (probably from incorrect versions of Ptolemy's work), ascribe to him the explanation of the fact, that the sun and moon on the horizon appear larger than in the zenith, as depending on the circumstance that, in the former case, our judgment is influenced by comparison with terrestrial objects. It has been ascertained by De Lambre, that in the original he does not give any thing like this explanation (which is, doubtless, the correct one), but only a very vague theory. This explanation is, therefore, probably due to his Arabic translator; and most likely derived from Alhazen.

meet.

Ptolemy distinguishes what has since been called the virtual focus, in the reflection from convex specula, or the point where the reflected rays, if produced, would And, among other interesting glimpses of truths not fully discovered till many ages afterwards, he notices the fact, that colours are confounded together by the rapidity of motion; and gives the instance of a wheel painted with different colours, and turned quickly round.

Progress of Mathematics.

Diophantus flourished at Alexandria in an early age of the Christian era; but his precise date is matter of question. He has left a treatise, in which the principles of what is in fact algebra, though not given in the form in which that science is now used, are laid down with great ingenuity and ability, in thirteen books, entitled "Arithmetical Questions." Some of the problems are of considerable intricacy; and much address is displayed in stating them, so as to bring out equations in a form involving only one power of the unknown quantity. The whole processes are expressed in common language, assisted by a very few symbols, which are mere abbreviations. Considering how little power and convenience the instrument had yet acquired with which he worked, it is remarkable how much he was able to effect. He directed his attention particularly to the class of problems called indeterminate, or which admit of a number of solutions: they have been hence known by the name of Diophantine problems, Though algebra was not at all yet reduced into a symbolical form, it is remarkable that Diophantus distinctly expresses the rule for the signs in multiplication, by saying, that λειψις into λειψις gives ύπαρξις, &c. (minus into minus gives plus).

The declining science of the fourth century was ably upheld by Pappus, one of the last distinguished ornaments of the Alexandrian school, who flourished between A. D. 350 and 400: he cultivated, with success, almost all branches of geometry; and, in most of them, has left some investigations of considerable value.

The subject of the loci was one to which his attention was especially directed; in particular, he considered that case which was called the "locus ad quatuor rectos;" a problem which baffled the powers of the

ancient geometry, but which has become celebrated as affording one of the highest triumphs to the modern analysis, and has led to the most extensive applications, of which we shall give an account in the sequel.

The principal work of Pappus is his "Collectiones Mathematica:" in this, among a variety of other curious discussions, we have a considerable space occupied in the establishment of the doctrine of maxima and minima, on geometrical grounds; and a remarkable application of it to the form of the cells of bees. This, in fact, involved the consideration of a class of problems which, in a more extended sense, have since been called isoperimetrical.* The nature of these will be rendered intelligible by a very simple illustration.-Let a circle be cut out in card, and its length of circumference measured by, and marked upon, a piece of tape wound round it; let then several other regular figures, as a triangle, square, hexagon, &c. be also cut out; by a little care in cutting, they may be gradually brought to such sizes, that the same tape will measure round their perimeters, so that they shall be all of exactly the same length as that of the circle; the perfect regularity of the respective figures being all the while carefully preserved. This being effected, it will be very easy to see, by merely placing them successively upon one another, that although their lengths of perimeter are exactly the same, yet their areas will be seen to be of palpably different extent.

To investigate such points by mathemetical reasoning is a matter of some difficulty; but the subject was treated by Pappus with great success in a variety of cases. It is found (and may be easily shown by the above method of illustration) that, of all figures having the same perimeter, that will have the greatest area which has the greatest number of sides: hence the circle (being regarded as a polygon of an infinite number of sides) will include the greatest area of all iso

perimetrical polygons. Similar considerations apply to solids and their surfaces.

The form in which the cells of the honeycomb are made appears to accord with the deductions of geometry, on principles of which this is the foundation, but to which several other considerations must be added. The only figures which, placed together, leave no interstices, are equilateral triangles, squares, and hexagons: of these, the last, of course, have the greatest content with the same circumference; in forming contiguous cells, therefore, there is a saving of materials in using the hexagonal form. The mode of terminating the cells is the most curious part of the whole. Geometrically, an extension of the same advantages would be gained if they ended in three-sided pyramids, formed by planes cutting off each alternate solid angle of the hexagonal cell, and inclined at a certain angle, which is found by calculation. Precisely such conditions have been found to be observed by taking the average of a great number of measurements of the actual cells of honeycombs.

This part of Pappus's work has come down to us in an imperfect state, but has been restored by Maclaurin. Pappus forcibly expresses his admiration at this very singular fact in the economy of the bee, exclaiming,* Κατα τινα γεωμετρικην μηχανώνται προνοιαν.” (They work by a sort of geometrical forethought.)

In his fourth book, Pappus discusses several points connected with the quadrature of the circle, and describes some of the inventions made by preceding geometers in their researches having that object in view. One of the most remarkable, perhaps, is the construction of a curve called the quadratrix, or squaring line, invented by Dinostratus and Nicomedes, of which he gives a demonstration; and by which the length of a circular arc is assigned: but the application of it involves the principle of limits, and does not, therefore, materially help the question. It affords, nevertheless, some very beautiful geometrical speculations.

The labours of this great geometer, however, extended