26 
martins aritfimetical frames. 
The form of Hie baloon will cause it to 
lie in the direction of the wind. If t!ie cur- 
rent of wind is only slightly contrary to the 
desired course, and the propeller.s can be 
made to act, tliere is little doubt, (he rudder 
H \Vil| enable tlie voyagers to keep their path, 
if the wind proves very contrary— and it is 
well known that the atmospheric currents 
are frequent and fitful— then the vovagers 
have no alternative but to descend a little,^ by 
which means an unfavorable, may often be 
changed for a favourable current. 'I’o effect 
partial descent when necessary, atmospheric 
air is forced into a small balloon inside of the 
large one, similar to the air bladder in fishes ; 
this can be filled or exhausted at pleasure by 
very simple means. 
If this small balloon is filled with atmos- 
pheric air, the gas in the large one will be 
compressed to such a degree, that, witli the 
load in the car, the whole machine will be 
rendered specifically heavier than the at- 
mosphere, and descend accordingly. On 
reaching a more favourable current, the crew 
withdraw the air from the small baloon, and 
the gas expanding restores a due proportion 
of the original buoyancy to the machine. 
This method may answer the pnrpo.se, but 
it appears to me that there is great risk of 
bursting the baloon, by compre.ssing the gas 
to such a degree as to effect efficient reduc- 
tion of the boyancy. A much better plan was 
proposed by Mr. G. th Atkinson, of New- 
castle-on-Tyne (in yonr 10th volume) viz. to 
withdraw a sufficient quantity of gas from the 
baloon by condensing it into a suitable cop- 
per ve.s.sel, and restoring it again to the bal- 
loon as required. 
With respect to tire inode of propulsion 
adopted by the inventors of the “Eagle,” I 
may just .state, that I do not consider it by 
any means the best that could be employed. 
1 remember hearing a lecture on aero- 
station delivered by lAlr. Tatum, some years 
since, in which he proposed to effect aerial 
navigation by means of two revolving \anes 
and a rudder. Avery considerable velocity 
could be imparted to a pair of vanes, without 
so great a loss of power as must necessarily 
take place in using wings. 
Balloons have for a long time past been 
mere toys, exhibited for the sake of gain ; and, 
1 confess, 1 am glad to see aerostation, as a 
science, is not entirely forgotten. There is, 
doubtless, but a very limited sphere of use- 
fulness 0 [)en to balloons ; but, 1 believe, much 
more can be accomplished than many persons 
are at present prepared to admit, I may re- 
turn to this subject again by-and-bye. In the 
mean time. 
I remain, yours respectfully, 
Wm. Baddelley. 
London, July 6,1835. 
MARTIN’S ARITHMETICAL FRAMES. 
Tlie powers of nnnrbers, and their relation 
to each other, have been in a variety of ways 
demonstrated ; but rarely indeed with any 
important practical application : we have in- 
genious theories of the wondrous powers of 
the number 9, and a variety of aritbmetical 
legerdemain is abroad, which appears to (he 
ciirion.s very singular and astoni.shing. Na- 
ftiei’s hones or rods afford some good illus- 
trations of the multiplying powers ; but there 
appears to have been no instance of the suc- 
cessful application of the “ occult powers of 
numbers” till the invention of the “ Arithme- 
tical Frames,” by Mr. Martin, which are, 
without question, applied to a use (be mo.'-t 
important and extensive. But wlien we come to 
niake an examination of these, we are unable 
to ascertain, except in one or two cases, the 
principles upon which they are constructed. 
1 u these frames we have \vhat is most ex- 
traordinary, a system of arrangement whii'ii 
carries out, ad infinitum, practical exhibi- 
tions ot all the elementary rules, not singly 
only, but also in every variety of combination 
which the ten digits will make, afi’ording de- 
monstrable proofs of the correctness or incor- 
rectne.ss of every figure ; at the same time that 
none hut the teacher «ho has been previously 
informed of (lie mode of detecting error, can 
by any possibility be informed of it. A dozen 
exercises of fifteen or sixteen figures each, 
may be worked in one rule only, or through 
the whole /oir?' rules, and be checked by the 
master at a mere glance, while those ex- 
ercises may be varied to the extent of 
many thousands of millions times, and 
be proved by the same mode and witlr 
tlte same facility. It has often occurred 
to mathematicians, that a series of num- 
bers might by some [.ossibility be arranged, 
so as to prod lice nnifonn and known results in 
an almost infinite series ; but this suspected 
power of the arrangement of numbers has 
never been sliown, excepting in a few cases 
of particular numbers ; and even these have 
not been applied (o any practical purpose, ex- 
ce[iting by Patiick Whytock.* But this ar- 
rangement, which is founded on the peculiar 
properties of certain decimal fractions is de- 
fective, as it only refers to the simple rules, 
wliereas the arithmetical frames or tablets 
constructed by iVir. Wartin, comprise also the 
compound rules ; and this appears most ex- 
traordinary ; for there cannot well be worse 
decimal lelatioiis, supposing they are con- 
structed on this principle, than (hose of the 
numbers 4, 12, and 20,whicl) form the integral 
parts of our common aurrency ; but Mr. 
Martin has arranged and can apply, if neces- 
sary, his principle through all the weights and 
measures , alFording an infinite variety of ex- 
amples, whose solutions bear such a relation 
to their propositions, that their correctness 
or incorrectness is immediately discoverable 
by him who has learnt the mode of discove- 
ry ; and which may be acquired, by any one 
conversent with addition and multiplication, 
in a few minutes. Nor is this all, for the 
frames are so arranged that the smallest as 
well as the largest examples may be given ; 
that the working of the examples, of one rule 
gives examples in another, and the workii^g 
again of these examples in a third, and so on 
— pi’oving the correctness of each, even 
to the pupils themselves, and pointing out 
* See Mech. Mag., vol. xviii p. 43. 
