486 

Here is a perfect demonstration evident to the senses. But let 
us go one step further. The rectangles in the preceding theorem 
may be bisected by diagonals and set round the square of the 
difference in such a manner as to form the square of the hypo- 
thenuse of the right-angled triangle, the sides of which are also 
those of the assumed squares. The squares of the sides of a right- 
angled triangle, therefore, are together equal to the square of the 
hypothenuse, since the former may be changed into the latter. 

The same conclusion may be arrived at by a still shorter and 
simpler course. Let any two squares be joined together as in 
the annexed figure, or, rather, let them be cut in paper in one 
piece. Then take ac equal to the side of the greater square, 
and joindcandce. Cut off the two equal triangles ac and 
cde, and place them in the positions of m/ and /7e, and the 
two squares will be thus transformed into the square of the 
hypothenuse of the right-angled triangle, of which they form the 
sides. 
Thus we have at oncea demonstration of the famous Pytha- 
gorean theoiem (Euclid i. 47), and have attained with three or 
four steps the same height climbed by Euclid with forty-seven. 
The words of his demonstration, committed to memory by a 
child, remain there mere words and nothing more. Words 
serve to mark and denote ideas, but cannot create them, where 
the material of ideas does not already exist. But the child who 
with paper or card amuses himself in going over the visible de- 
monstration suggested in this letter, in various forms and re- 
peatedly—for neither old nor young can be said to learn a truth 
merely by its transient recognition —will assuredly awaken to an 
agreeable consciousness of the reasoning faculty, and feel no 
difficulty in future geometrical studies. 
In 1860 there was published for me, by Messrs. Williams and 
Norgate, a little volume entitled, ‘‘ The Elements of Geometry 
Simplified and Explained,” adapted to the system of empirical 
proof, and of exhibiting the truth of theorems by means of 
figures cut in paper. It contained in thirty-five theorems the quint- 
essence of Euclid’s first six books, together with a supplement of 
thirty-three not in Euclid. There was no gap in the sequence or 
chain of reasoning, yet the 32nd and 47th provisions of Euclid 
were respectively the 3rd and 17th of my series. This book 
proved a failure, for which several reasons might be given, but it 
will be sufficient here to state but one, namely, that it came forth 
ten years before its time. What became of it I know not. But of 
this Iam convinced, that though I failed, success will attend 
those who follow in my footsteps. W. D. CooLry 
THE discussion in your last part on methods of teaching 
elementary geometry reminds me that at a period when I was 
teaching the subject to a considerable number of pupils, I fre- 
quently overcame the difficulties of very young or inapt students 
by commencing with the study of a so/id, such as a cube, en- 
couraging the pupils to frame definitions for the parts of the 
object. The ideas existing in the child’s mind of a solid, a plane, 
a line, and a point, were thus put into words in an order the 
reverse of that in which they would have been had Euclid been 
used. The chief properties of parallelograms and triangles fol- 
lowed, and were easily discovered by the use of a pair of com- 
passes, scissors, and paper, and that at an age when Euclid was 
a sealed book. I believe children can be most easily taught to 
solve problems in plane geometry when they occur in connection 
with early instruction in practical solid geometry. Most children 
try to draw, and if they were encouraged to represent simple 
objects by “‘ plans” and ‘‘ elevations,” the necessity of obtaining 
aknowledge of how to describe the forms presented to them 
would frequently carry the pupils through a large number of the 
principal problems of plane geometry with a pleasure they could 
not experience if the ‘* problems” were put before them, without 
any reason for their solution but the teacher’s command. The 
powers of truthful representation gained by such teaching, would 
NATURE 



[Oc¢. 19, 1871 

be of the utmost value to thousands who would never attempt to 
learn ‘‘ Euclid ;” whilst, so faras I am able to judge, it is more 
likely to prepare the boy to read formal works on geometry with 
pleasure than to create a distaste for the study, 
THOMAS JONES 
Woolwich, October 9 
The Coming Eclipse 
I HAVE been very much interested in Mr. Lockyer’s lecture 
at the Royal Institution on the lateeclipse. I am especially glad 
that he is at length able to acknowledge the existence of com- 
paratively cool hydrogen, because in my Eclipse Report of 
1868 (vol. xxxvii. Part 1, R.A.S, Memoirs), I stated that I 
believed from the evidence of the photographs that hydro- 
gen was dispersed from the prominences in visible streams in 
some cases, and in others invisibly. 
But while Mr. Lockyer admits this, he seems to me very un- 
necessarily to avoid everywhere the use of our familiar term 
** atmosphere” to include the whole gaseous envelope of the 
sun, This seems to me to be the sense in which Kirchhoff used 
the word when he said it was extensive.* It certainly was the 
sense in which I used it, and, I believe, that in which all who 
spoke of an extensive atmosphere did so use it. In this sense 
there can be no doubt that the sun Zas an extensive atmosphere, 
the outer portion of which is comparatively cold and capable of 
reflecting light if the polarisation now not doubted be due to 
reflection. 
There is one consideration which, however, does not seem to 
have occurred to Mr. Lockyer. If the cold atmosphere, as I 
will venture still to call it, reflect the prominence light, it will 
also reflect the solar light. Its reflected light then should be 
such as reaches us at ordinary times, and not so exclusively 
chromospheric. Adding to this the light which is due to cool 
| hydrogen, we should have, I anticipate, a faint continuous spec- 
trum with the bright line F, and also a solar spectrum with, 
perhaps, some of the chromospheric lines reversed. That is not 
what has been found, and I do not at present see any way of 
reconciling the facts with the theory that the undoubted polarisa- 
tion is due to reflection. 
Before going to another subject, I would wish to direct atten- 
tion to my friend Captain (now Major) Branfill’s observations in 
1868+ on the polarisation of the corona. Mr. Proctor, indeed, 
in his book on the Sun, says that the Astronomer Royal did not 
consider them conclusive, but I have his official statement that 
he did so consider them, and an inquiry as to Mr. Proctor’s 
authority leads me to think that Mr. Airy’s meaning was mis- 
taken. I think any one who reads the account in the original 
will feel that the plane of polarisation was satisfactorily deter- 
mined. An observer in 1870 has said that he found the bands 
of Savart persistent. I have not now time to look up the 
reference, but he used, it seemed to me, the centre of the moon 
as the centre of rotation. Captain Branfill was careful not to 
do this, as his figures prove (page 25 of Report). 
Now to the future. I have received from Government an in- 
quiry as to recommendations to observers coming out. I am 
now suggesting, in addition to my own station at Dodabetta, that 
observers should be stationed at Kotagherry in the Nilgherries, 
at Manantoddy among the coffee districts to the west, and at 
Tirupur, close to Avenashy Road Station of the Madras Railway. 
Of these Manantoddy is the least accessible, but the whole will 
give a range of stations from $8,600 feet high down to the 
ordinary level of the plain country. More observers could be 
accommodated on the Nilghernes, where the weather, I am 
assured, is likely to be excellent. Of Ceylon I have not satis- 
factory accounts, nor of the west coast. 
If these stations be adopted, I would suggest that, if possible, 
there should be a conference of observers. The possibility will 
depend on our leisure, which, probably, none of us can now 
oresee, 
I should say that I have made these suggestions without 
reference to Mr. Pogson, because I know nothing of his plans, 
having received no answer to inquiries ; it is possible those may 
modify projects, but any visitors should bear in mind that it is 
almost necessary that some European residences should be close 
to their stations. J. T. TENNANT 
H.M, Mint, Calcutta, Sept. 11 
* Mr. Lockyer has long ago shown that the Sun’s atmosphere lies partly 
above and partly below the superior limit of the photosphere. The word 
Atmosphere was used by Kirchhoff in the manner indicated, because he 
believed the photosphere to be liquid.— Eb. 
+ American observers seem neyer to have seen the Report. 
