Oct. 12, 1871 | 
NATURE 
465 

of the third-book method, I can readily show that such an infer- 
ence was by no means so absurd as might be inferred from 
his remarks about the oldness of the problem. For I 
remember distinctly an occasion on which the solution of 
this problem was required during a lecture at King’s College, 
London, at which my friends Baily (second wrangler in 
1860) and Hudson (third wrangler in 1861) were present, 
Three of the students at once submitted to the lecturer 
the solution by the sixth-book method (which no one can well 
miss), and the lecturer (a second wrangler), while admitting that 
the solution was not very pleasing, was unable at the moment to 
suggest a better ; he added, jokingly, that the best way to solve 
the problem would be to describe a parabola having the given 
point as focus and either of the given lines as directrix. Now, 
he had not been long engaged in teaching, and it may be per- 
fectly true that one who had been so engaged ‘‘ would certainly 
have in his memory one or more solutions of this problem ;” but 
this would depend on the subjects he had been engaged upon. 
If he chinced to be one of the most eminent mathematical 
professors at Cambridge, it is probable that no problem in the 
higher analysis would be unknown to him, but the odds would 
be rather against than in favour of his being familiar with the 
best solutions of geometrical problems, just as the odds would 
be against his being proficient in the rules for ‘‘ Barter,” ‘‘ Tare 
and Trett,” and ‘* Alligation Partial.” 
From letters which have reached me I find that the general 
purport of my letter has been misapprehended, since some 
appear to infer that I question the geometrical power of our 
University mathematicians. I meant nothing so unreasonable. 
We have geometricians who rival (and I believe more than rival) 
in power the best Continental geometricians. But their geome- 
trical strength has not been attained during their University 
career ; and no one who considers carefully the mathematical 
course at either University, can believe that it tends either to form 
geometricians or to foster geometrical taste. 
I candidly admit that 1 do not speak of either course from 
personal experience. All I know of geometry was learned before 
my Cambridge time, and very nearly all I know of analytical 
mithematics was learned after that time. But I know quite well 
the nature of each course, and-can sustain my statement that our 
universities do not encourage the study of geometry. Whether 
they should do so is a matter on which I have expressed no 
Opinion, RICHD. A, PROCTOR 
Brighton, Oct. 7 
P.S.—Mr. Todhunter refers to the actual solution of the 
problem as a ‘‘ matter of some interest, though of course uncon- 
nected with the theoretical solution.” As I have had some ex- 
perience in constructive geometry (having always made it a practice 
to solve astronomical problems constructively betore proceeding 
to numerical calculation), I may be permitted to make some 
remarks on this point. First I would add to the compasses and 
parallel ruler (the only instruments mentioned by Mr. Todhunter) 
that most useful instrument the square. With this instrument 
(which would be needed in any case) the following construction 
would be as convenient as the one founded on the sixth-book 
solution. The problem, be it remembered, requires that a circle 
should be described through a given point to touch two given 
straight lines. Let P be the point, AB and AFCG the lines, 
AHDK the bisector of BAC (this bisector must be drawn in both 
methods, so that I leave its construction untouched) ; with the 
square draw CPD square to AK, and PE square to CD ; with 
centre D draw circular arc PE ; with centre C and distance PE 
draw half circle FLG ; then FH and GK drawn square to AG 
(with the square) are radii of the two circles fulfilling the con- 
ditions, 
Prof. Newcomb and Mr. Stone 
In Mr. Proctor’s letter to NATURE of the 23rd ult., he 
remarks that Prof. Newcomb had stated to him that he was 
bewildered at having a discussion of the transits of Venus and 
the parallax of the Sun, deducible from them, prior to that of 
Mr. Stone, attributed to himself; and Mr. Proctor goes on to 
state that he was justified in his belief that such a discussion 
had been made because a writer, signing himself “P. S.” had 
asserted that it had in a letter appearing in the Astronomical 
Register for December 1868. He further gives two reasons for 
the unhesitating credit which he had given to the assertion of 
*©P.S.” The first of them is that there is strong internal evi- 
dence that the writer was a distinguished astronomer having 
those as his initials (or a part of his initials, it would be more 

correct to say); of this it seems scarcely needful to say more, as 
the writer in question may prefer not to be unearthed. But of Mr. 
Proctor’s other reason, may J be permitted to say a word? It 
is that the assertion of ‘‘P.S,” was ‘‘ permitted to remain un- 
corrected.” 
Had Mr. Proctor turned to the very next number of the 
Astronomical Register (that for January 1869) he would have 
found a letter signed also with initials ‘‘ W. T. L.” in which 
“*P. S.’s”? assertion that Prof. Newcomb had published any dis- 
cussion of the transit of Venus in 1769, is most emphatically 
contradicted. ‘‘P.S.,” it is true, made a rejoinder in the March 
number of the Register (page 65) but in it he neither denies 
“W. T. L.’s” contradiction, nor refers (as of course he could not) 
to any original investigation of the transit-of-Venus problem by 
Prof. Newcomb. He contented himself with the rather unin- 
telligible remark that ‘‘ W. T. L.’s” answer was not in ‘‘the 
spirit of the age we live in.” The latter writer in the following 
number of the Register (page 88) pointed how, in all probability, 
the mistake of ‘*P.S.”’ had arisen from misunderstanding part 
of the title of a paper by Prof. Newcomb on the Distance of the 
Sun, and the matter dropped. 
Now, as Prof. Newcomb was as likely to have seen 
‘“W. T. L.’s” contradiction as ‘‘ P. S.’s” assertion, there would 
certainly seem no necessity for his further disowning himself 
what ‘‘ P. S.” had claimed for him. W. T. LYNN 
Blackheath, Oct. 2 
Note on the Cycloid 
I po not know whether it has been noticed that the cycloid is 
a projection of the common helix (thread-inclined 45°). I sup- 
pose the property must have long since been recognised, but have 
not seen it mentioned. 
The proof is very simple, and may be thus presented :— 
Suppose a vertical circle to have its plane east and west (a lu- 
minous point, for the nonce), the sun in the meridian and 45° high. 
Then the shadow of the circle on a horizontal plane will clearly 
be a circle ; and further, if a point move uniformly round the 
vertical circle, the shadow of the point will move uniformly round 
the shadow-circle. Now, let the centre of the vertical circle 
advance horizontally towards the south, while a point moves 
round its circumference at the same uniform rate. The moving 
point will describe a right helix with a thread-inclination of 45% 
Its shadow will move uniformly round the shadow-circle while 
the centre of this circle advances uniformly and at the same rate 
in a straight line. It will therefore describe a cycloid. 
It is obvious that all the varieties of curtate and prolate-cycloids 
may be obtained as projections of helices, by changing the thread- 
inclination. 
Also it is obvious that if the sun (or the point of projection) 
were in the zenith, the shadow (or projection) of the helix first 
dealt with would be the curve called ‘‘the companion to the 
cycloid.” RicHarp A, PRrocror 
Is Blue a Primary Colour ? 
In recent works on colour blue is called a primary colour. If 
blue is a primary colour a mixture of yellow and blue tran- 
sparent pigments could not produce green, but would form an 
opaque combination. The colour produced by a mixture of 
yellow and blue pigments—if blue is an elementary colour— 
will depend on the colour reflected by the coloured layer itself, 
and not on the light passed through it from the white surface 
underneath. The brilliancy of the green produced by mixing 
yellow and blue pigment, isa measure of the transparency, to 
the green rays, of the blue pigment employed. Or in other 
words, there is as much green in the blue pigment employed, as 
there is green in the green produced by mixing that pigment 
with yellow. Blue must, therefore, be a compound colour, since 
the blue pigment passes the green rays. 
Further. When the light reflected from blue substances is 
examined with a prism, it is found to be composed of green and 
violet. Again, when green and violet are combined by means 
of a rotating disc, blue is produced. By varying the proportions 
of green and violet any colour from green through blue-green or 
sea-green, blue, blue violet or indigo to violet, may be obtained, 
Again, when the solar spectrum is thrown on a blue surface, the 
green and the violet rays are reflected in the same way as a 
yellow surface reflects the red and the green rays. 
The following is a simple way of showing that blue is not an 
elementary colour, and that violet is an elementary colour :— 
Take a piece of red, a piece of green, and a piece of violet 
