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To know from this the form of this spheroid, that is to say the proportion of the semi-diameters CS, CP, of its elliptical section, which are perpendicular to one another, I considered that the point M where the ellipse is touched by the straight line FH, parallel to CG, ought to be so situated that CM makes with the perpendicular CL an angle of 6 degrees 40 minutes; since, this being so, this ellipse satisfies what has been said about the refraction of the ray perpendicular to the surface CG, which is inclined to the perpendicular CL by the same angle.
But how does the case really stand? Copernicus and Tycho Brahe held the distance to be twelve hundred semi-diameters; Kepler, who is received to have been perhaps the greatest astronomer that any age has produced, puts it down as three thousand five hundred semi-diameters; since his time, Riccioli as seven thousand; Hevelius as five thousand two hundred and fifty ; some later astronomers, mentioned by Halley, as fourteen thousand; and Halley himself as sixteen thousand five hundred .
For the Ellipse HME is given, and its conjugate semi-diameters are CH and CM; because a straight line drawn through M, parallel to HE, touches the Ellipse HME, as follows from the fact that a plane taken through M, and parallel to the plane HDE, touches the spheroid at that point M, as is seen from Articles 27 and 23.
"The mean distance of the moon from the earth, in the syzygies in semi-diameters of the earth, is, according to Ptolemy and most astronomers, 59; according to Vendelin and Huygens, 60; to Copernicus, 60 1/3; to Street, 60 2/3; and to Tycho, 56 1/2. Correct this error and the distance will become about 60 1/2 semi-diameters of the earth, near to what others have assigned.
Common readers are only informed of the latest adjustment of the question, and are therefore unavoidably led to believe that the distance of the sun from the earth, ever since astronomy became entitled to the name of a science, has by universal consent been recognised as ninety-five millions of miles, or, as near as may be, twenty-four thousand semi-diameters of the earth.
For the versed sine of that arc which the moon, in the space of one minute of time, would by its mean motion describe at the distance of sixty semi-diameters of the earth, is nearly 15 1/12 Paris feet, or more accurately 15 feet, 1 inch, 1 line 4/9.
The power that two wheels of different magnitudes have upon each other is in the same proportion as if the semi-diameter of the two wheels were joined together and made into that kind of lever I have described, suspended at the part where the semi-diameters join; for the two wheels, scientifically considered, are no other than the two circles generated by the motion of the compound lever.
If one considers further the other pieces H of the wave AC, it appears that they will not only have reached the surface AB by straight lines HK parallel to CB, but that in addition they will have generated in the transparent air, from the centres K, K, K, particular spherical waves, represented here by circumferences the semi-diameters of which are equal to KM, that is to say to the continuations of HK as far as the line BG parallel to AC. But all these circumferences have as a common tangent the straight line BN, namely the same which is drawn from B as a tangent to the first of the circles, of which A is the centre, and AN the semi-diameter equal to BC, as is easy to see.
This wave is then represented by the circumference SNR, the centre of which is A, and its semi-diameter equal to two-thirds of CB. Then if one considers in order the other pieces H of the wave AC, it appears that in the same time that the piece C reaches B they will not only have arrived at the surface AB along the straight lines HK parallel to CB, but that, in addition, they will have generated in the diaphanous substance from the centres K, partial waves, represented here by circumferences the semi-diameters of which are equal to two-thirds of the lines KM, that is to say, to two-thirds of the prolongations of HK down to the straight line BG; for these semi-diameters would have been equal to entire lengths of KM if the two transparent substances had been of the same penetrability.
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