FEBRUARY 3, 1923] 
NATURE 
153 


had not discovered the modern developments of 
certain functions as series; nor had they discovered 
that differentiation and integration are the inverse 
of one another. There is little trace in Greek geo- 
metry of considerations corresponding to the differential 
calculus; the only case that seems certain is that 
of the subtangent property of a spiral which must 
have been obtained by the consideration of the in- 
stantaneous direction of motion, at any point on the 
curve, of the point describing it. If, as is probable, 
Apollonius, in his treatise on the cochlias or cylindrical 
helix, dealt with tangents to the curve, he would no 
doubt determine the direction of the tangent at any 
point in the same way. 
But the Greeks were by no means limited to what 
they could obtain by direct integration. They were 
very ingenious in reducing an integration which they 
could not perform directly to another the result of 
which was already known. This must have been 
the method by means of which Dionysodorus found 
the content of an anchor-ring or tore and Pappus 
obtained his theorem which anticipated what is known 
as Guldin’s theorem. In the matter of the anchor- 

ring the Greeks also anticipated Kepler’s idea that the 
coritent is the same as if the ring be conceived to be 
straightened out and so to become a cylinder. The 
Method of Archimedes is mostly devoted to the reduc- 
tion of one integration to another the result of which 
is known, but is remarkable also as showing how he 
obtained certain results otherwise proved in his main 
treatises. The method was a mechanical one of 
measuring elements of one figure against elements 
of another, the elements being expressed as parallel 
straight lines in the case of areas and parallel plane 
sections in the case of solids. This point of view 
anticipated Cavalieri. The elements are: really in- 
finitesimals, indefinitely narrow strips and indefinitely 
thin laminze respectively, though Archimedes does 
not say so. But Archimedes disarms any criticism 
that could charge him with using infinitesimals for 
proving propositions by carefully explaining that the 
mechanical procedure does not constitute a proof 
and is only useful as indicating the results, which 
must then, before they are definitely accepted, be 
proved by geometrical methods, that is, by the method 
of exhaustion. 
The Disappearing Gap in the Spectrum.! 
By Prof. O. W. RicuHarpson, F.R.S. 
Il, 
ee NG to Fig. 1, B, which is repeated here for 
7 convenience of reference, this shows the various 
outposts where from time to timespectral lines have been 
located. It will be seen that there is still a con- 
siderable gap between 16-35, the limit obtained with 
the vacuum grating at the L series of aluminium, and 
S908 
A INFRA Red VisiBic] Quare 
WVARTZ Rire 
is z 4 6 8 
1s 
17°39, the limit with the crystal spectrometer at the 
L series of zinc. Between these limits no spectral 
lines are known, but there is evidence of the excitation 
of such lines, and data have been obtained for the high- 
frequency limits of spectra in this region. 
This evidence depends upon considerations of a 
somewhat different character from those dealt with 
1 Continued from p. rar. 
NO. 2779, VOL. 111] 
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so far. If we consider any typical characteristic X- 
radiation of an element, for example, the K-radiation, 
it is found to consist of a number of spectral lines which 
are denoted by the symbols Ka, Kg, EK, in order of 
ascending frequency. In general there are more than 
three lines, but we shall adopt the symbol K, for the 
line of highest frequency which is observed, and we 

shall denote its frequency by Vey: These lines can be 
an element by a stream either of high- 
or of high-velo¢ ity electrons 
reaching it. In either case the lines are not excited 
separately, but the whole group K,-K, appears 
simultaneously. It is found that there are simple and 
important restrictions on the radiation frequencies 
and on the electron energies which are capable of 
excited in 
frequency radiation 
