"At its higher levels the golden age of Muslim civilization was both an immense scientific success and a exceptional revival of ancient philosophy. These were not its only triumphs... but they eclipse the rest. ...[T]he Saracens ...made the most original contributions [to science]. These, in brief, were nothing less than trigonometry and algebra... In trigonometry the Muslims invented the sine and the tangent. The Greeks had measured an angle only from the chord of the arc it subtended: the sine was half the chord. The Chosranian (...Mohammed Ibn-Musa) published in 820 an algebraic treatise which went as far as quadratic equations: translated into Latin in the sixteenth century, it became a primer for the West. Later, Muslim mathematicians resolved biquadratic equations. Equally distinguished were Islam's mathematical geographers, its astronomical observatories and instruments (in particular the ) and its excellent if still imperfect measurements of and , correcting the flagrant errors of Ptolemy."