454 Comparison between the Chords of Arcs 
Institution, in which it has been twice carefully distilled at a low 
heat, it has still contained minute quantities of phosphorus. 
It has been supposed that dry phosphoric acid is fixed at a 
white heat ; but I find that this is not the case: it rapidly rises 
in vapour at this temperature, and evaporates even at the point 
of fusion of flint glass: and the hydrate of phosphoric acid is 
susceptible of being volatilized at a much lower temperature. 
In converting the solid sublimate composed of phosphorus , 
and chlorine into the liquid compound, When the phosphorus is 
first used in contact with the sublimate, a yellow crystalline mass 
is formed, which, when acted on by a higher degree of heat, 
affords the liquid chloride, which rises from it in vapour, and 
leaves phosphorus behind. It is possible that this yellow solid 
is a compound of phosphorus and chlorine, containing half asmuch. 
chlorine as the liquid. Should this be proved to be the case by 
future exeriments, it will give weight to the idea, that the hypo- 
phosphorous acid is a binary compound of oxygen and phosphorus. 
LXVIII. Comparison between the Chords of Arcs employed by 
ProLemy and ihose now in Use. By G.A.WALKER ARNOTT, 
A.M. Edinburgh. 
To Mr. Tilloch. 
Sir, — Ix a late elegant publication (The Philosophy of Arith- 
metic), it is stated that the ratio of 1 to3-1416, or of the dia- 
meter to the circumference of a circle, must have been almost 
known to Claudius Ptolemy. This celebrated philosopher and 
mathematician, and first of ancient astronomers, left behind 
him, in the third book of his Almagest, a table of the chords 
of the arcs of the circle, calculated in sexagesimals to every 30’ 
or half-degree, and which are found to coincide with those in 
the trigonometrical tables we at present employ, with a much 
more considerable degree of exactness than could reasonably be 
looked for from the small advances made at that time in this sub- 
ject. It is therefore my purpose here to exhibit a table of com- 
parison between these, the insertion of which in your Magazine 
may gratify such of the curious as may not have seen the work 
itself of this distinguished man. 
The first column contains the chords of every two degrees of 
the semicircle, as calculated by Ptolemy in sexagesimals. In 
the second column are the same chords converted into sexagesi- 
mals from our common decimal tables: and here I may add, 
that that number is taken, nearest to which, either above or be- 
low, the true number approaches, when extended to thirds, 
fourths, &c. of the radius.. In the third are Ptolemy’s calcula- 
tions turned into decimals; and in the fowrth we have an ex- 
tract 
