begin with fastening two sheets together lengthwise. The edges of two
sheets are laid down so as to lap or cover each other an inch, and a
slip of the same copper, about three and a half inches broad, called
the reeve, is introduced between them. Four oblong holes, or slits,
are then cut or punched through the whole, and they are fastened or
riveted together by copper nails, with small round shanks and flat
heads. Indents are then cut 1¾ inch deep upon the seam at top and
bottom. The right hand sheet and the reeve are then folded back to
the left. The reeve is then folded to the right, and the sheets being
laid on the roof in their place, it is nailed down to the sheeting
with flat-headed short copper nails. The right hand sheet is then
folded over the reeve to the right, and the whole beat down flat
upon the cartridge paper covering the sheeting, and thus they are
fastened and laid in their places, by nailing down the reeve only;
and by reason of the oblong holes through them and the reeve, have a
little liberty to expand or contract with the heat and cold, without
raising themselves up from the sheeting, or tearing themselves or
the fastening to pieces. Two other sheets are then fixed together,
according to the first and second operations above, and their seam,
with the reeve, introduced under the upper ends of the seam of the
former, so as to cover down about two inches upon the upper ends
of the former sheets: and so far the cartridge paper is allowed
to cover the two first sheets. This edge of the paper is dipped in
oil, or in turpentine, so far before its application, and thus a
body between the sheets is formed impenetrable to wet, and the reeve
belonging to the two last sheets is nailed down to the sheeting as
before, and the left hand sheet is turned down to the right. Four
sheets are now laid down, with the seam or joint rising to the ridge;
and thus the work is continued, both vertically and horizontally,
till the roof be covered, the sides and ends of each sheet being
alternately each way, undermost and uppermost.
The price for copper, nails, and workmanship, runs at about eight
pounds ten shillings per hundred weight, or two shillings and
three-pence per foot, superficial, exclusive of the lappings; and
about two shillings and eight-pence per foot upon the whole; which
is rather above half as much more as the price of doing it well with
lead.
TO PETER COLLINSON, ESQ. AT LONDON.
_Magical Square of Squares._
SIR,
According to your request I now send you the arithmetical curiosity,
of which this is the history.
Being one day in the country, at the house of our common friend, the
late learned Mr. Logan, he showed me a folio French book filled with
magic squares, wrote, if I forget not, by one M. Frenicle, in which
he said the author had discovered great ingenuity and dexterity in
the management of numbers; and though several other foreigners had
distinguished themselves in the same way, he did not recollect that
any one Englishman had done any thing of the kind remarkable.
I said, it was, perhaps, a mark of the good sense of our English
mathematicians, that they would not spend their time in things that
were merely _difficiles nugæ_, incapable of any useful application.
He answered, that many of the arithmetical or mathematical questions,
publicly proposed and answered in England, were equally trifling and
useless. Perhaps the considering and answering such questions, I
replied, may not be altogether useless, if it produces by practice an
habitual readiness and exactness in mathematical disquisitions, which
readiness may, on many occasions, be of real use. In the same way,
says he, may the making of these squares be of use. I then confessed
to him, that in my younger days, having once some leisure (which I
still think I might have employed more usefully) I had amused myself
in making these kind of magic squares, and, at length, had acquired
such a knack at it, that I could fill the cells of any magic square
of reasonable size, with a series of numbers as fast as I could
write them, disposed in such a manner as that the sums of every row,
horizontal, perpendicular, or diagonal, should be equal; but not
being satisfied with these, which I looked on as common and easy
things, I had imposed on myself more difficult tasks, and succeeded
in making other magic squares, with a variety of properties, and
much more curious. He then shewed me several in the same book, of an
uncommon and more curious kind; but as I thought none of them equal
to some I remembered to have made, he desired me to let him see
them; and accordingly, the next time I visited him, I carried him a
square of 8, which I found among my old papers, and which I will now
give you, with an account of its properties. (_See Plate_ V. Fig. 3.)
The properties are,
1. That every strait row (horizontal or vertical) of 8 numbers added
together makes 260, and half each row half 260.
2. That the bent row of 8 numbers, ascending and descending
diagonally, viz. from 16 ascending to 10, and from 23 descending
to 17; and every one of its parallel bent rows of 8 numbers make
260.--Also the bent row from 52 descending to 54, and from 43
ascending to 45; and every one of its parallel bent rows of 8 numbers
make 260.--Also the bent row from 45 to 43, descending to the left,
and from 23 to 17, descending to the right, and every one of its
parallel bent rows of 8 numbers, make 260.--Also the bent row from
52 to 54, descending to the right, and from 10 to 16, descending to
the left, and every one of its parallel bent rows of 8 numbers make
260.--Also the parallel bent rows next to the above-mentioned, which
are shortened to 3 numbers ascending, and 3 descending, &c. as from
53 to 4 ascending, and from 29 to 44 descending, make, with the two
corner numbers, 260.--Also the 2 numbers 14, 61 ascending, and 36,
19 descending, with the lower 4 numbers situated like them, viz. 50,
1, descending, and 32, 47, ascending, make 260.--And, lastly, the 4
corner numbers, with the 4 middle numbers, make 260.
[Illustration:
_Plate XI._ _Vol. II. page 327._
_A Magic Square of Squares._
_Published as the Act directs, April 1, 1806, by Longman, Hurst, Rees
& Orme, Paternoster Row._]
So this magical square seems perfect in its kind. But these are not
all its properties; there are 5 other curious ones, which, at some
other time, I will explain to you.
Mr. Logan then shewed me an old arithmetical book, in quarto, wrote,
I think, by one Stifelius, which contained a square of 16, that he
said he should imagine must have been a work of great labour; but if
I forget not, it had only the common properties of making the same
sum, viz. 2056, in every row, horizontal, vertical, and diagonal.
Not willing to be out-done by Mr. Stifelius, even in the size of my
square, I went home, and made, that evening, the following magical
square of 16, which, besides having all the properties of the
foregoing square of 8, _i. e._ it would make the 2056 in all the same
rows and diagonals, had this added, that a four-square hole being cut
in a piece of paper of such a size as to take in and show through it
just 16 of the little squares, when laid on the greater square, the
sum of the 16 numbers so appearing through the hole, wherever it was
placed on the greater square, should likewise make 2056. This I sent
to our friend the next morning, who, after some days, sent it back
in a letter with these words: "I return to thee thy astonishing or
most stupendous piece of the magical square, in which----" but the
compliment is too extravagant, and therefore, for his sake, as well
as my own, I ought not to repeat it. Nor is it necessary; for I make
no question but you will readily allow this square of 16 to be the
most magically magical of any magic square ever made by any magician.
(_See the Plate._)
I did not, however, end with squares, but composed also a magic
circle, consisting of 8 concentric circles, and 8 radial rows, filled
with a series of numbers from 12 to 75 inclusive, so disposed as that
the numbers of each circle, or each radial row, being added to the
central number 12, they make exactly 360, the number of degrees in
a circle; and this circle had, moreover, all the properties of the
square of 8. If you desire it, I will send it; but at present, I
believe, you have enough on this subject. I am, &c.
B. FRANKLIN.
TO THE SAME.
_Magical Circle._
SIR,
I am glad the perusal of the magical squares afforded you any
amusement. I now send you the magical circle. (_See Plate_ XII.)
Its properties, besides those mentioned in my former, are these.
Half the number in any radial row, added with half the central
number, make 180, equal to the number of degrees in a semi-circle.
Also half the numbers in any one of the concentric circles, taken
either above or below the horizontal double line, with half the
central number, make 180.
And if any four adjoining numbers, standing nearly in a square, be
taken from any part, and added with half the central number, they
make 180.
[Illustration:
_Plate XII._ _Vol. II. page 328._
A MAGIC CIRCLE OF CIRCLES.
_Published as the Act directs, April 1, 1806, by Longman, Hurst, Rees
& Orme, Paternoster Row._]
There are, moreover, included four other sets of circular spaces,
excentric with respect to the first, each of these sets containing
five spaces. The centres of the circles that bound them, are at A, B,
C, and D. Each set, for the more easy distinguishing them from the
first, are drawn with a different coloured ink, red, blue, green,
and yellow.[62]
These sets of excentric circular spaces intersect those of the
concentric, and each other; and yet the numbers contained in each of
the twenty excentric spaces, taken all around, make, with the central
number, the same sum as those in each of the 8 concentric, viz. 360.
The halves, also of those drawn from the centres A and C, taken above
or below the double horizontal line, and of those drawn from centres
B and D, taken to the right or left of the vertical line, do, with
half the central number, make just 180.
It may be observed, that there is not one of the numbers but what
belongs at least to two of the different circular spaces; some to
three, some to four, some to five; and yet they are all so placed as
never to break the required number 360, in any of the 28 circular
spaces within the primitive circle.
These interwoven circles make so perplexed an appearance, that it
is not easy for the eye to trace every circle of numbers one would
examine, through all the maze of circles intersected by it; but if
you fix one foot of the compasses in either of the centres, and
extend the other to any number in the circle you would examine
belonging to that centre, the moving foot will point the others out,
by passing round over all the numbers of that circle successively. I
am, &c.
B. FRANKLIN.
FOOTNOTE:
[62] In the plate they are distinguished by dashed or dotted lines,
as different as the engraver could well make them.
TO THE REV. FATHER BECCARIA.
_Describing a new musical Instrument composed of Glasses._
_London, July 13, 1762._
REV. SIR,
I once promised myself the pleasure of seeing you at Turin, but as
that is not now likely to happen, being just about returning to my
native country, America, I sit down to take leave of you (among
others of my European friends that I cannot see) by writing.
I thank you for the honourable mention you have so frequently made
of me in your letters to Mr. Collinson and others, for the generous
defence you undertook and executed with so much success, of my
electrical opinions; and for the valuable present you have made me
of your new work, from which I have received great information and
pleasure. I wish I could in return entertain you with any thing new
of mine on that subject; but I have not lately pursued it. Nor do I
know of any one here that is at present much engaged in it.
Perhaps, however, it may be agreeable to you, as you live in a
musical country, to have an account of the new instrument lately
added here to the great number that charming science was before
possessed of.--As it is an instrument that seems peculiarly adapted
to Italian music, especially that of the soft and plaintive kind,
I will endeavour to give you such a description of it, and of the
manner of constructing it, that you or any of your friends, may
be enabled to imitate it, if you incline so to do, without being
at the expence and trouble of the many experiments I have made in
endeavouring to bring it to its present perfection.
You have doubtless heard the sweet tone that is drawn from a drinking
glass, by passing a wet finger round its brim. One Mr. Puckeridge, a
gentleman from Ireland, was the first who thought of playing tunes,
formed of these tones. He collected a number of glasses of different
sizes, fixed them near each other on a table, and tuned them by
putting into them water more or less, as each note required. The
tones were brought out by passing his fingers round their brims.--He
was unfortunately burned here, with his instrument, in a fire which
consumed the house he lived in. Mr. E. Delaval, a most ingenious
member of our Royal Society, made one in imitation of it, with a
better choice and form of glasses, which was the first I saw or
heard. Being charmed with the sweetness of its tones, and the music
he produced from it, I wished only to see the glasses disposed in a
more convenient form, and brought together in a narrower compass, so
as to admit of a greater number of tones, and all within reach of
hand to a person sitting before the instrument, which I accomplished,
after various intermediate trials, and less commodious forms, both of
glasses and construction, in the following manner.
[Illustration: (hemispherical glass vessel)]
The glasses are blown as near as possible in the form of hemispheres,
having each an open neck or socket in the middle. The thickness of
the glass near the brim about a tenth of an inch, or hardly quite so
much, but thicker as it comes nearer the neck, which in the largest
glasses is about an inch deep, and an inch and half wide within,
these dimensions lessening as the glasses themselves diminish in
size, except that the neck of the smallest ought not to be shorter
than half an inch.--The largest glass is nine inches diameter, and
the smallest three inches. Between these three are twenty-three
different sizes, differing from each other a quarter of an inch in
diameter.--To make a single instrument there should be at least six
glasses blown of each size; and out of this number one may probably
pick thirty-seven glasses (which are sufficient for three octaves
with all the semitones) that will be each either the note one wants
or a little sharper than that note, and all fitting so well into each
other as to taper pretty regularly from the largest to the smallest.
It is true there are not thirty-seven sizes, but it often happens
that two of the same size differ a note or half note in tone, by
reason of a difference in thickness, and these may be placed one in
the other without sensibly hurting the regularity of the taper form.
The glasses being chosen, and every one marked with a diamond the
note you intend it for, they are to be tuned by diminishing the
thickness of those that are too sharp. This is done by grinding them
round from the neck towards the brim, the breadth of one or two
inches, as may be required; often trying the glass by a well tuned
harpsichord, comparing the tone drawn from the glass by your finger,
with the note you want, as sounded by that string of the harpsichord.
When you come nearer the matter, be careful to wipe the glass clean
and dry before each trial, because the tone is something flatter when
the glass is wet, than it will be when dry;--and grinding a very
little between each trial, you will thereby tune to great exactness.
The more care is necessary in this, because if you go below your
required tone, there is no sharpening it again but by grinding
somewhat off the brim, which will afterwards require polishing, and
thus increase the trouble.
The glasses being thus tuned, you are to be provided with a case
for them, and a spindle on which they are to be fixed. My case is
about three feet long, eleven inches every way wide within at the
biggest end, and five inches at the smallest end; for it tapers all
the way, to adapt it better to the conical figure of the set of
glasses. This case opens in the middle of its height, and the upper
part turns up by hinges fixed behind. The spindle, which is of hard
iron, lies horizontally from end to end of the box within, exactly
in the middle, and is made to turn on brass gudgeons at each end.
It is round, an inch diameter at the thickest end, and tapering to
a quarter of an inch at the smallest.--A square shank comes from
its thickest end through the box, on which shank a wheel is fixed
by a screw. This wheel serves as a fly to make the motion equable,
when the spindle, with the glasses, is turned by the foot like a
spinning-wheel. My wheel is of mahogany, eighteen inches diameter,
and pretty thick, so as to conceal near its circumference about 25lb
of lead.--An ivory pin is fixed in the face of this wheel, and about
four inches from the axis. Over the neck of this pin is put the loop
of the string that comes up from the moveable step to give it motion.
The case stands on a neat frame with four legs.
To fix the glasses on the spindle, a cork is first to be fitted in
each neck pretty tight, and projecting a little without the neck,
that the neck of one may not touch the inside of another when put
together, for that would make a jarring.--These corks are to be
perforated with holes of different diameters, so as to suit that
part of the spindle on which they are to be fixed. When a glass is
put on, by holding it stiffly between both hands, while another turns
the spindle, it may be gradually brought to its place. But care must
be taken that the hole be not too small, lest in forcing it up the
neck should split; nor too large, lest the glass not being firmly
fixed should turn or move on the spindle, so as to touch and jar
against its neighbouring glass. The glasses thus are placed one in
another, the largest on the biggest end of the spindle which is to
the left hand; the neck of this glass is towards the wheel, and the
next goes into it in the same position, only about an inch of its
brim appearing beyond the brim of the first; thus proceeding, every
glass when fixed shews about an inch of its brim (or three quarters
of an inch, or half an inch, as they grow smaller) beyond the brim
of the glass that contains it; and it is from these exposed parts of
each glass that the tone is drawn, by laying a finger upon one of
them as the spindle and glasses turn round.
My largest glass is G, a little below the reach of a common voice,
and my highest G, including three compleat octaves.--To distinguish
the glasses the more readily to the eye, I have painted the apparent
parts of the glasses within side, every semitone white, and the other
notes of the octave with the seven prismatic colours, _viz._ C, red;
D, orange; E, yellow; F, green; G, blue; A, indigo; B, purple; and C,
red again;--so that glasses of the same colour (the white excepted)
are always octaves to each other.
This instrument is played upon, by sitting before the middle of the
set of glasses as before the keys of a harpsichord, turning them
with the foot, and wetting them now and then with a spunge and clean
water. The fingers should be first a little soaked in water, and
quite free from all greasiness; a little fine chalk upon them is
sometimes useful, to make them catch the glass and bring out the tone
more readily. Both hands are used, by which means different parts are
played together.--Observe, that the tones are best drawn out when the
glasses turn _from_ the ends of the fingers, not when they turn _to_
them.
The advantages of this instrument are, that its tones are
incomparably sweet beyond those of any other; that they may be
swelled and softened at pleasure by stronger or weaker pressures of
the finger, and continued to any length; and that the instrument,
being once well tuned, never again wants tuning.
In honour of your musical language, I have borrowed from it the name
of this instrument, calling it the Armonica.
With great esteem and respect, I am, &c.
B. FRANKLIN.
TO A FRIEND[63].
_Respecting the best Mediums for conveying Sound._
_July 20, 1762._
DEAR SIR,
I have perused your paper on sound, and would freely mention to
you, as you desire it, every thing that appeared to me to need
correction:--But nothing of that kind occurs to me, unless it be,
where you speak of the air as "the _best_ medium for conveying
sound." Perhaps this is speaking rather too positively, if there be,
as I think there are, some other mediums that will convey it farther
and more readily.--It is a well-known experiment, that the scratching
of a pin at one end of a long piece of timber, may be heard by an
ear applied near the other end, though it could not be heard at the
same distance through the air.--And two stones being struck smartly
together under water, the stroke may be heard at a greater distance
by an ear also placed under water, than it can be heard through the
air. I think I have heard it near a mile; how much farther it may be
heard I know not; but suppose a great deal farther, because the sound
did not seem faint, as if at a distance, like distant sounds through
air, but smart and strong, and as if present just at the ear.--I wish
you would repeat these experiments now you are upon the subject, and
add your own observations.--And if you were to repeat, with your
naturally exact attention and observation, the common experiment of
the bell in the exhausted receiver, possibly something new may occur
to you, in considering,
1. Whether the experiment is not ambiguous; _i. e._ whether the
gradual exhausting of the air, as it creates an increasing difference
of pressure on the outside, may not occasion in the glass a
difficulty of vibrating, that renders it less fit to communicate to
the air without, the vibrations that strike it from within; and the
diminution of the sound arise from this cause, rather than from the
diminution of the air?
2. Whether, as the particles of air themselves are at a distance
from each other, there must not be some medium between them, proper
for conveying sound, since otherwise it would stop at the first
particle?
3. Whether the great difference we experience in hearing sounds at
a distance, when the wind blows towards us from the sonorous body,
or towards that from us, can be well accounted for by adding to or
subtracting from the swiftness of sound, the degree of swiftness that
is in the wind at the time? The latter is so small in proportion,
that it seems as if it could scarce produce any sensible effect,
and yet the difference is very great. Does not this give some hint,
as if there might be a subtle fluid, the conductor of sound, which
moves at different times in different directions over the surface of
the earth, and whose motion may perhaps be much swifter than that of
the air in our strongest winds; and that in passing through air, it
may communicate that motion to the air which we call wind, though a
motion in no degree so swift as its own?
4. It is somewhere related, that a pistol fired on the top of an
exceeding high mountain, made a noise like thunder in the valleys
below. Perhaps this fact is not exactly related: but if it is, would
not one imagine from it, that the rarer the air, the greater sound
might be produced in it from the same cause?
5. Those balls of fire which are sometimes seen passing over a
country, computed by philosophers to be often thirty miles high at
least, sometimes burst at that height; the air must be exceeding rare
there, and yet the explosion produces a sound that is heard at that
distance, and for seventy miles round on the surface of the earth,
so violent too as to shake buildings, and give an apprehension of an
earthquake. Does not this look as if a rare atmosphere, almost a
vacuum, was no bad conductor of sound?
I have not made up my own mind on these points, and only mention them
for your consideration, knowing that every subject is the better for
your handling it.
With the greatest esteem, I am, &c.
B. FRANKLIN.
FOOTNOTE:
[63] Mr. Oliver Neave. _Editor._
TO LORD KAIMS, AT EDINBURGH.
_On the Harmony and Melody of the old Scotch Tunes._
_June 2, 1765._