This is a transcription of the writings of John 'Longitude' Harrison which
are related to his concept of "** Radical Numbers**".(a
method of calculating the proportions to be used in the manufacture of bells)
and were found in his "lost" manuscript "

To hear how this pi-derived tuning sounds, when used in lullabies, and see a related flash cartoon go here. (C.E.H.L. March 2005)

Start of relevant transcription:

Let not any Thing as hereafter [viz. as about the proportioning of bell] as
from the beginning of this section be printed; not but that the Rules thereto
are right, was the Radical Numbers, as thereunto appertaining to be rightly
acquir'd. Now my next, or last thing is to shew or treat of the Rules which
I discover'd for calculating the proportions for Peals of Bells, i.e. so far
as appertains, or may appartain to their diameters at the Skirt, and Thicknesses
at the Sound-Bow; but it is not hence to be understood (as already noify'd)
but that there must be a Tuning, but (to the purpose) that less may do. [i.e.
when they are withal, from other due proportions and care so founded] as well
as thence at the same time to be able to continue with as it were undisturbed
Uniformity as touching their Courage, I say, as with respect to these good properties,
that the less Chipping may do: And as Peals of Bells are only Musical Instruments
we have in the World that will stand in Tune for a long time, i.e. supporting
they be, or can be by any accurate means so render'd at first, and as that as
above can now be the Case [tho' not so before my Completion of the Monochord]

+++ Start of Footnote:

A Pipe (as is to be understood) being nothing to the purpose of Tuning, if it
even could be made upon the true Foundation, but that is what it cannot be,
without (as at second hand) the Assistance of a Monochord, no, nor with it neither
[viz. to come to sufficient perfection] notwithstanding its being at the same
time to be blown with Bellows, as I well know from Experience.

End of Footnote++++

Such an attainment therefore , at least as when upon a proper, or the most proper
Foundation [as the which I am now about to describe] must be worth the while
to be done, so inconsequence of which I shall proceed, wishing withal that the
Ringers would always use 'em in so becoming a manner as they ought to do, and
that is, besides good raising and Gearing [and that as much the best with the
Bells rightly hung, as well as when at the same time in due proportion] never
to practice that ridiculous Absurdity of Changing the Tenor. So first, It must
be very improper for the Tenor Bell in any Peal to be too poor or thin, for
a poor Tenor [or Cover?] makes a poor Peal, but that it ought to be as light
as it well can for its bigness to be good: whence [as from Experience] it is
necessary that the Thickness at the Sound-Bow of the Tenor Bell, when no more
Bells than Eight in the Peal be about 1/15 of its Diameter at the Skirt; but
when for a Tenor in Ten or Twelve, about 1/16 of the same do better.Now I have
found, that towards the right proportioning of Bells, there must be a Number
assign'd so as fitly to be Denominated the Radical Number Correspondent to the
Diameter (at the Skirt) of some one Tenor Bell, and from thence it might be,
that the Radical Number corresponding to the Diameter of any other Tenor Bell
of what bigness so ever is to be found; and the same is to be observ'd as with
regard to the Thickness, there being also to be a Radical Number assign'd for
that purpose, viz to the Thickness (at the Sound-Bow) of some one [and the same]
Tenor bell as before from whence the radical Number to the Thickness of the
other Tenor Bell is likewise to be found, but by different means as followeth.

The Radical Numbers corresponding to the Diameter at the Skirt of tenor Bells
in different Peals, must be (accordingly as I have discover'd) in a sub duplicate
Ratio of the said Diameters; and the radical Numbers corresponding to the Thickness
of the same at the Sound-Bows, in a duplicate Ratio of the said Thickness.

Thus, Let it be suppos'd as from Experience [notwithstanding the want of Confirmation
from some proper Essays or Experiments] that the Radical Number Correspondent
to the Diameter of a Tenor of 52 Inches must be 13624727367, and whence the
Radical Number corresponding to the Diameter of any other Tenor, as suppose
of 37 Inches is to be found; do therefore, by the following Analogy say,

As the square Root of 52, viz. 7,2111025 is to this given [or as suppos'd] Radical
Number, viz. 13684727367: So is the Square Root of 37, viz 6,08276253 to 1154344
the other radical Number corresponding.

++ Start of Footnote

Not that five or six of these last Figures in the first radical Number, or as
here especially in this first [estimated] radical Number are at all material
[since as hereafter the whole is or is to be us'd as a Decimal Fraction] save
only for the instancing therein for finding the Logarithm of the Number [ or
of a Number of so many places] as by and by, the which as hereafter is useful,
but still, not meaning it as absolutely necessary (in this Affair) ever to have
so many places, save only as by way of Curiosity.

End of Footnote +++

And as with respect to the Thickness at the Sound-Bow; the 15th part of 52 is
3,4667 - and the square of which is 12,01766222, so according to the former
supposing, let 1476926182 be suppos'd the corresponding Radical Number to this
Thickness and then say,

As the Square of 3,4667 - viz. 12,01766222 is to this Radical Number viz. 14769262:
so is the square of the 15th part of 37, i.e. the square of 2,4667 - 6,08436222
to 0747745605 the Radical Number sought.

But Note, when in finding the radical Numbers as this by or from their corresponding
Square Roots and squares, the whole of each Radical Number, as already exprest,
is then to be taken or look'd upon as a Decimal Fraction, and must likewise
be so esteem'd when in subtracting one from another as in the Operations below;
and as so in finding the said Radical Numbers it will be fairly perceivable
that a Cypher must sometimes be prefix'd, as is the case in the last radical
Number here above; but afterwards, when in finding the Logarithm of such a Radical
Number, that of the first five figures; but when such a Cypher, the Characteristick
of the Logarithm thereunto belonging must be 3, 4 being assign'd when the first
Figure is not a Cypher.

Example for finding the Logarithm of the Radical Number 13684727367

The Log. found for the first five Figures is 4.1362130655

230838,645854

__________________

4,1362361494

Then for the residue 9 as followeth, and) as here added say,

As an Unite with so many Cyphers as there ate Figures remaining, is to the same
Figures [as they stand]:

Page 127:

so is the difference betwixt the Log. found and that which follows to another
Number;

That is, As 10000000 is to 727367; so is 317362 to 230838,645854; the which
being added to the Log. found, as in order here above gives the Log. required,
viz. 4,1362361494.

Examples for finding the corresponding Radical Number to a given Logarithm.

Suppose the given Log. to be 4,1362361494

Find the Log. next less, as being the Log. of the first five Figures in the
Radical Number sought, the which will be the Log. of 13684, and the which Log.
itself is 4,1362130655

So their Difference is --------------------------- 230839

Then divide this difference by the Difference betwixt this Log. next ???les,
and that which follows, annexing the Figures as they come out to the first five
Figures of the Absolute Number, will give the Radical Number sought,Thus 317362)
2308390 ( 13684727368

2221534

___________

868560

634724

__________

2338360

2221534

_______________

1168260

952086

____________

2161740

1904172

___________

2575680

2538896

______________

36784

Now this Operation is the same as to say, As 317326 is to 230839; so is 100000000
to 727368 the six last Figures.

Now the last Figure here, from reason to be perceiv'd above, not being exactly
what it ought to be is not material.Now the Foundation being laid. the next
Things to be notify'd are the Logarithms to be us'd in order for obtaining the
?ust Dimensions (viz. so far as above spoken to) of all the Bells many Peal
design'd and they are, nay, must certainly be according to the same proportion
as them, which, as I have said, according to the secret course of nature (as
in the former part of this Book} naturally results from the circumference, Diameter,
and Radius of a Circle, for, as must each one [viz. accordingly as in tha respect
to the Foundation laid] be but one Quarter of its Correspondent or Natural one,
and them for the Thicknesses Double, so they are in each respect as in the two
following Tables.

Log., viz. as deduc'd from the former part of this Book 1/4 Log. for the Diameters.

Octave ----------- ,0752575

5th --------------- ,0436175

4th --------------- ,03164

Sharp 3rd. -------- ,023955

Larger Note ------- ,0119775

Lesser Note ------- ,007685

LogsLog.'s, or as according to the former part of this Book double Logs. for
the Thicknesses.

Octave ----------- ,60206

5th ------------- ,34894

4th -------------- ,35312

Sharp 3rd -------- ,19164

larger Note ------ ,09582

Lesser Note ------ ,06148Things being now as farther prepar'd, I come to their
use as with regard to the right, or due proportioning of Peals of bells; viz.
so far as touching the Diameters at their Skirts, and Thickness at their Sound-Bows.

And first, as with regard to the Diameters, let there be a Line drawn down the
middle of the Space in which the Operation is to be wrought [for the Examples
below], and then on the right hand side thereof, and close to the Line, let
the Logarithm of the Diameter of the Tenor Bell [as taken from a Book or Table
of Logarithms] be plac'd; and at some distance towards the right hand, and along
the same height, put the said Diameter itself:

And on the left hand side the Line, and still at the same height or level, place
the Radical Number; then (as still towards the left) prefix the Logarithm corresponding
to the Radical Number; and then to begin the Work, Take one of the Logs for
the Diameter from the Table above, as Suppose that of the larger Note, and place
it under, and subtract it from the Logarithm last above mention'd,

+++Start of Footnote:

see below, in the lower part of the first Example.

End of Footnote+++

and the absolute Number corresponding to the remainder [or as then to that remaining
Logarithm] place as a Radical Number under the other, and subtract it therefrom,
and put their Difference on the right hand side the Line under the Log. of the
Diameter, but with this remark, that each Figure respectively be plac'd one
place nearer the Comma than it falls to be [i.e. in order that it may be render'd
10 times as big as it is] then substract it (so plac'd) from the Log. of the
Diameter, and the Remainder or Difference will be the Log. of the Diameter there
sought, and so on, proceeding again with the Log. of the larger Note as in the
Example. And the same Method as here shewn is to be observed for the Thickness,
save only without the increase of the Differences of the Radical Numbers, i.e.
there without the placing 'em a Figure more forward or to the multiple of 10;
the Examples will make all plain to such as understand the Logarithms.

If another Operation be wrought, and that as first if you please, and as therein
making use of the Log. for the Octave, or rather of the Logs. for the 5th and
4th, it will serve as a Proof to the truth of the Work; Example 1st at the end
of the Book.

Now, from a pretty deal of pains or labour, this together with that for the
corresponding Thickness, Example 2nd were the Only Rules, which, not only had
beauty in Proportion in themselves, but the which I could discover that would
at the same time turn out the Bells in Tune, or as more suitably speaking [viz.
according to what will commonly happen] nearly so, for it is with supporting,
not only the Metal to be uniform throughout the whole peal, but that the Bells
could be moulded, as not only so far as with respect thereto, but as farther
in such other respects as may result therefrom relating to their shape or proportion
upwards, each one in itself to a Mathematical degree of exactness, and this
must be also with supporting in the first place these Radical Numbers, viz.
as corresponding to a tenor Bell of 52 Inches Diameter to be rightly acquir'd,
the which as imply'd above is hard to say, since before I imagin'd there might
be, or hit upon the use of such Numbers, there never was for this purpose any
thought about such in the World, much less, as now at first to be expected their
Mathematical Acquisition, no in such a case, experience must rectify or confirm:
And , as according to which, my Nephew Henry harrison, viz. some years after
my drawing up of the Papers, told me from Experience, that my Radical Numbers
were not right [no surely, to have been so would have been a Wonder indeed]
but that my Rule or Rules were good, and that he had [by Experience] acquir'd
the real and secret radical Numbers:

But what I am here more chiefly about to observe is, that from these Rules barely,
i.e. without any thing derivation as below, the trebles in heavy Peals will
be lighter in proportion, and consequently weaker in Courage as with respect
to the tenors, or as with respect to what they ought to be, than them in lighter
Peals, altho as with respect to Tune, as imply'd above the same, and indeed
from the random methods or guesses of the Bell-Founders, such weakness for the
most part part has hitherto been a general fault, but as there can be no reason
why it should be so [as even contradicted by themselves, viz. by their chance
methods or guesses in lighter Peals and as I shall manifestly shew] but that
on the other hand, and both as primarily with respect to properness of Courage,
as well as secondarily with regard to Ponderosity, and as augmented by describing
such largeness of Circle, as may from their hanging be coincident thereto, and
that in order for the fineness in raising and Gearing,

+++++Start of Footnote:

Or fineness of Raising and Gearing from the nature of Pendulums.

End of Footnote +++

and as not withal forgetting Chiming, as well as for the Ringing of Changes,
the Treble in Eight ought always [or in all Peals] to be in Diameter about 2/3
of that of the Tenor or perhaps not quite so much, or but about as 5/8; hence
the following Derivation, as a farther Step to these Rules, or to this Rule
must be necessary. At the bottom of the Operation, as with respect to the Diameters,
Examples 1st., is to be notify'd, that from the Log. of 34.6666666 &c. i.e.
from the Log. of 2/3 of 52 is substracted the Log. of somewhere about 31,497,
and their Difference (as there) being added to the last Radical Number shews
what that last Radical Number ought to have been viz. 11923814774 and consequently
its corresponding Log. 4,0764152208, and these in order to have turn'd out the
Diameter of the treble 2/3 of that of the Tenor; and as from this Log. there
is farther substracted the Log. of the last radical Number, shews the portion
wanting as with respect to the Log. of the said last Radical Number to have
compleated the same, i.e. to have turn'd out the Treble (in Diameter) 2/3 of
that of the Tenor; but what is the chief matter to be observ'd or gather'd from
this last mention'd substraction is , that by or from its remainder is shewn
the portion wanting, as taken in proportion to the Log. of the larger note to
compleat the Diameter of the treble, and not only so, but from thence as a Foundation
is to be ascertain'd the Diameters and Thicknesses of all the intermediate Bells
as well as them of the treble, viz. when the said Treble [as herein Eight] is
to be in Diameter 2/3 of that of the Tenor [or as always meaning what it ought
to be ]; so towards the Matter by the following Analogies say,

And first, as on course with respect to the Diameters, Example 1st.

As the Log. of the larger Note there, viz. ,0119775 is to 1 [i.e. to the Note
itself]; so is the last remainder there, or as here above spoken of viz. ,0154365714
to 1,288797.

And then with respect to the Thickness, Example 2nd

As 1 is to ,09582 [the Log. of the larger Note there]; so is 1,288797 to ,12349252854

2

_____________

The double of which is ----------------------------------- ,24698505708

Now from this Log. [as here doubled] being added to the Log. of the last Radical
Number [as with respect to the Thickness, and as at the bottom of Example 2nd]
is produc'd, viz. by first substracting the radical Number next above, from
the Radical Number correspondent to this Log. so increas'd and then by their
Difference being added to the Log. of the last Thickness, is produc'd ( i say)
the Log, for the Thickness of the Treble when its Diameter is to be 2/3 of that
of the Tenor; and the reason why this last Quotient [or Logarithm as here above]
must be doubled is obvious as may be perceived from the Calculations, for supposing
the Diameter of the 2nd bell was taken to be equal to that of the 3rd Bell,
and still to retain its Note [or Pitch] as the 2nd, it must then in Thickness
be equal to that of the 4th; and the same may be said of the 6th Bell, for was
its Diameter to be the same as the 7th, its Thickness must then be the same
as the 8th.

And now, According to what has been advanc'd;

The Log. of the last Thickness being ,4291426201 The Thickness itself is 2,6862

The Difference of the radical Number is ,0282822887 So the Thickness sought
is 2,867

Whence 2,867 must be the Thickness of the treble, when its Diameter is 2/3 of
that of the tenor of 52.

And now; As the Diameter of the Treble is to be 34.6666&c. [i.e. 2/3 of
52], and as its Thickness is then to be 2,867, so a new radical Number in each
respect, or new radical Numbers must be had for the tenor wherefrom to begin
the calculations, and they must each one be a good deal less respectively than
what they were before: As to the Radical Number for finding the diameter [at
all times] was it not as hath been shewn, or as fairly appears, viz. that the
others from the Square Roots (as above) must, as in the first place be us'd,
in order for finding a Consequent Foundation for the Thickness,

+++Start of Footnote:

But it may not be amiss to observe, that one Operation as with the Log. of the
Octave would or will be sufficient for the purpose (tho' in example 1st. is
taken in all the intermediate Bells), not but that the same one Operation [so
far as I had conceived of the matter] must have requir'd several trials as in
the Note below, had there not been a farther Discovery made.

End Of Footnote +++

I say was it not so, one and the same Radical Number would always do, without
any more to do for the Diameters whether in great or small Peals, and that radical
Number would be, or rather is 1106771141, for from thence [and as with Beauty
In Proportion throughout the Peal] the diameter of the treble in Eight will
always be turn'd out exactly n2/3 of that of the Tenor:

+++ Start of Footnote:

Nay from thence consequently, as the Differences betwixt the radical numbers
respectively must always [or in all Peals] continue the same, so therefore each
one of which being substracted from its correspondent Log. for the Diameter,
must always produce what is wanting or sought for [viz. without having recourse
to the Log. of the Radical Numbers, or then of the larger and lesser Notes],
as for Ignorance, the difference betwixt the radical Number of the tenor-Bell
and that of the bell next to it, being always then ,0301068075, the which being
substracted from the Log. of the Diameter of the Tenor [be it what bigness soever]
will always leave the Log. of the Diameter of the next Bell, but this cannot
be the Case as touching the Thickness; but however, for finding the Foundation
for the said Thicknesses from that of the Diameters, there can be no necessity
thence but only to have one Operation, or as indeed will commonly [in this point
of the Matter] prove repeated Operations [was there not the famous discovery
by Mr. Charlton as below] and that from alterations in the Radical Number, viz.
in that which must be approximated for the thickness of the tenor, and consequently
in its Logarithm as thereunto corresponding, beginning at first [viz. towards
finding the said Radical Number] with an approximation of three or four of the
first Figures, and then [viz. after each such alterations] with the Log. of
the Octave only, 'till the Log. of the Thickness requir'd for the Treble be
rightly turn'd out; nay it may be observ'd, that was it not with respect to
a Foundation for the Thicknesses, that the Diameter, after their first attainment
for some one Peal [as here for the Tenor of 52] might thence be had by the Golden
Rule" or Lute? or Rate? alone, viz. as implying the Diameter of the Treble
in Eight to be always 2/3 of that of the Tenor.

End of Footnote:+++But the Radical Number for the Thicknesses [was it not again
as discover'd below] would always, or as in consequence have been to have been
found by making Essays with the Log. of the Octave [viz. as after each such
alteration as hath just been spoken of in the Note above] and they might sometimes
as there shewn have prov'd a pretty many before it could be hit upon; now the
last of which Essays I made for my purpose here, and wherein I found the Number
to be truly acquir'd, I have put for Example at the top of the calculations
for the thicknesses Example 4th; but was it not to be so nicely ascertain'd
it would still do, since four or five of the last Figures in the Log.'s of the
thicknesses as in that of the treble &c.. [when so many] will not sensibly
affect the matter: And the same ( as said above) may be observ'd as touching
the Diameters:

But as to the finding the new Radical Number for the Thicknesses viz. after
the Thickness of the Treble in Eight is at any Time, or for any design'd Peal
as above ascertain'd, mr. Lionell Charlton, teacher of the Mathematicks at Whitby
in Yorkshire, did, after the Communication of my papers to him , as by my nephew
Mr. henry harrison of Barrow, did I pay luckily, or as more properly through
sagacity hit upon a method for finding the radical Number for the Tenor, viz.
the radical Number, to a Mathematical truth, wherefrom to begin the calculations
for the Thicknesses of all intermediate Bells - without repeated Operations
as spoken of above, and the which is as followeth. Let the Logarithm of the
Thickness of the treble in Eight, be substracted from the Logarithm of the Thickness
of the Tenor,

Now since the Differences of the radical Numbers for the readier finding the
respective diameters in or for any peal [as intimated in the Note last above]
do not appear any farther, or can only be had from the following Calculations
respectively as for the Diameters in a peal of Eight; therefore I have here
inserted other four Differences, viz. as touching the Diameters of the first
four Bells in a Peal of twelve, the which would have been seen from the said
Calculations, had they been continued up so far:

These two to mark up Ten ,0253167065

,0246280319

And these other two to make up Twelve,0154478504

,0235378726Let the calculations, with the Similar view of the Sizes, and Thicknesses
therefrom follow here.

Now from the calculations it may withal, be gather'd that the Thickness of the
treble, when in Diameter 2/3 of that of the tenor is not in proportion so thick
as when of less Diameter; for from the said Calculations may be had the following
Analogy,

Viz. As 31,497 is to 2,6862: so 34,666 &c. to 2,9565

Whenas the Thickness of the Treble is but 2,867;

So this Treble is in proportion to its Diameter thinner than the other by ,0895,
consequently not to be clogg'd with metal, whatever has been, or may be alledg'd
from the mismanagement or blundering of the Bell-Founders; and in consequence
of which most of the Peals hitherto, but especially the large ones, are pretty
much short in goodness or laudableness of what they ought to be, or might have
been, i.e. as primarily touching the uniformity and pleasantness of their Courage;
as well as besides the great Matter wanting, viz. as with respect to their being
in Tune, save only in this latter point, a few Peals to be excepted the which
have been tun'd by my Monochord: so hence I think I may safely pronounce that
any Bell-Founder whatever, who, besides his Honesty in making use of good metal,
can hereafter come up to these Rules of Art, as principally here described,
and as together with what may therefrom or thereby, be farther acquir'd as touching
the Thicknesses, or Proportions of Bells upwards, and as moreover with the properness
of descent from the middle of the Sound-Bow to the Skirt, the which, as notifying
the out-side of the Bells, must be more towards uprights in the treble, than
in the tenor-Bell, and that because the treble is thicker in proportion to its
bigness; I say, any Bell-Founder, who in all respects can hereafter turn off
what may be esteemed as truly Laudable, ought certainly to be encourag'd for
his Labour: And (supposing the Ringers to forsake, or to be occasion'd to forsake
their Rascally nonsense in Changing the Tenor} it is to be wish'd that our gentry
could take some thought towards what might be done this way [as affording some
Adoration towards the Publick Worship of Almighty God, or as in other words
bearing some relation to the chief Drift or Contents of this Book, i.e. to what
ought to be] and less, or as might in consequence prove much less towards that
of the Charms of the Play-House &c. and withal, as the same viz. the chief
design or matter of this Book is to be supported at far less Expence than the
said Play-House &c., then as so consequently, much less towards the thoughts
of extravagancy in raising their Rents, or as otherwise in consequence towards
the oppressing or grinding the Poor or the most industrious part of the Nation,
and the which they themselves could not subsist without to Powder, and so consequently,
according to the Royal Psalmist [as in psalm 144, with the 84] more towards
the good or happiness of the Nation, as pronouncing them Blessed, who long to
tread this other Path, or have delight therein [ and as thence withal to keep
themselves out of danger of what is pronounc'd in the 82 Psalm] a difficult
matter now indeed! as being, by one means or other almost become an estrang'd
or rather as it were a forfeiting Thing, far short sure in Vertue, goodness
or worthiness of King David's Designs and dedications, or otherwise he must
have been instead of a Type of Christ, a very weak Man! whereas what he said
appointed and asserted was not over-set, put out of date or fashion by Christ,
neither as in consequence ought it so to be by the Ministry of the Gospel [
as by any up set new-fashion'd Preacher] no nor certainly by any other means
whatever; consequently, the main Drift of this Book cannot but stand good in
itself, whatever Use it may be of, or Abuse in lieu be put upon it, and this
must be my Conclusion.