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 "A true and full account of the foundation of Musick, or, as, principally therein, of the Existence of the Natural Notes of Melody" of circa 1776.

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)

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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]
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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.
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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.
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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.
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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
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]:
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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
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,
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see below, in the lower part of the first Example.
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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,
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Or fineness of Raising and Gearing from the nature of Pendulums.
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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
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,
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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.
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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:
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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
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.

Modern drawing with list of bell elements

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