Trigonometry
Trigonometry did not take its rise from
the general consideration of the periodicity of nature. In this respect its history is analogous to that of conic sections, which also had their origin in very particular ideas. Indeed, a comparison of the histories of the two sciences yields some very instructive analogies and contrasts. Trigonometry, like conic sections, had its origin among the Greeks. Its inventor was Hipparchus (born about 160 ),
a Greek astronomer, who made his observations at Rhodes. His services to astronomy were very great, and it left his
hands a truly scientific subject with important results established, and the right method of progress indicated. Perhaps the invention of trigonometry was not the least of these services to the main science of his study. The next man who extended trigonometry was Ptolemy, the great Alexandrian astronomer,
whom we have already mentioned. We now
see at once the great contrast between conic sections and trigonometry. The origin of trigonometry was practical; it was invented because it was necessary for astronomical research. The origin of conic sections was purely theoretical. The only reason for its initial study was the abstract interest of the ideas involved. Characteristically enough conic sections were invented about years earlier than trigonometry, during the very best period of Greek thought. But the importance of trigonometry, both to the theory and the application of mathematics, is only one of innumerable instances of the fruitful ideas which the general science has gained from its practical applications.
We will try and make clear to ourselves what trigonometry is, and why it should be generated by the scientific study of astronomy.
In the first place: What are the measurements which can be made by an astronomer? They are measurements of time and measurements of angles. The astronomer may adjust a telescope (for it is easier to discuss the familiar instrument of modern astronomers) so that it can only turn about a fixed axis pointing east and west; the result is that the telescope can only point to the south, with a greater or less elevation of direction, or, if turned round beyond the zenith, point to the north. This is the transit instrument, the
great instrument for the exact measurement of the times at which stars are due south or due north. But indirectly this instrument measures angles. For when the time elapsed between the transits of two stars has been noted, by the assumption of the uniform rotation of the earth, we obtain the angle through which the earth has turned in that period of time. Again, by other instruments, the angle between two stars can be directly measured. For if is the eye of the astronomer, 22 and and are the directions in which the stars are seen, it is easy to devise instruments which shall measure the angle . Hence, when the astronomer is forming a survey of the heavens, he is, in fact, measuring angles so as to fix the relative directions of the stars and planets at any instant. Again, in the analogous problem of