section
Observationes Cyclometricæ
by Adam Adamandy Kochański – Latin text with annotated English translation
translated by Henryk Fukś
Department of Mathematics, Brock University, St. Catharines, ON, Canada
email hfuks@brocku.ca
Translator’s note: The Latin text of Observationes presented here closely follows the original text published in Acta Eruditorum [2]. Punctuation, capitalization, and mathematical notation have been preserved. Several misprints which appeared in the original are also reproduced unchanged, but with a footnote indicating correction. Every effort has been made to preserve the layout of original tables. The translation is as faithful as possible, often literal, and it is mainly intended to be of help to those who wish to study the original Latin text.
1 \pstart
ADAMI ADAMANDI E SOCIET. JESU
Kochanski Dobrinniaci, Sereniss. Poloniarum Regis Mathematici & Bibliothecari, OBSERVATIONES Cyclometricæ,
ad facilitandam Praxin accomodatæ;
ex Epistola ad Actorum Collectores.
\pend
Qui Mathemata serio coluerit, nec tamen ad difficillima quæque & adhuc insoluta Problemata vires ingenii sui pertentandas censuerit, vix quenquam repertum esse existimo. Haud equidem diffiteor, me quoque olim eodem morbo laborasse, & ut alia præteream, in Circulo quidem quadrando, vel examinandis aliorum in eo conatibus, operæ non nihil collocasse. Non attinet hic enumerare Methodos, quas ea in re secutus fueram: unam tantum, quam fortasse quispiam felicius excolere poterit, commemorabo. Persuaseram mihi conjectura quadam, possibiles esse aliquas Rectarum sectiones, quarum segmenta invicem, & cum aliis rectis Longitudine vel Potentia incommensurabilia essent, Circuli tamen Areæ, vel Peripheriæ partibus Longitudine aut Potentia commensurarentur; ita ut inventa sectione istiusmodi, liceret ex ea Tetragonismum expedire Geometrice, vel saltem rationem Diametri ad Ambitum, in numeris ad lubitum maximis supputare. \pend
Ad eam porro cogitationem videbar mihi non temere, sed illius Quadratricis, a Dinostrato inventæ, ductu devenisse. At cum ab istis laboribus ad alia disparata studia animus avocaretur, illum tandem adjeci, & quidem magnorum Virorum exemplis incitatus, ad investiganda compendia quædam Cyclometrica, Praxibus mechanicis utilia, idque tam in Numeris, quam Lineis; quorum nonnulla hoc loco adferre lubet. \pend
1 \pstart
BY ADAM ADAMANY FROM THE SOCIETY OF JESUS
Kochański of Dobrzyń^{1}^{1}1Dobrzyń nad Wisłą – Kochański’s birthplace, a town in Poland on the Vistula River, with settlement history dating back to 1065., Mathematician and Librarian of the Most Serene King^{2}^{2}2John III Sobieski (1629 – 1696), from 1674 until his death King of Poland and Grand Duke of Lithuania. of Poland,
Cyclometric OBSERVATIONS,
accommodated for easiness of practical use; from a letter to fellow readers of Acta.
I suppose one could hardy find anyone who would seriously cultivate knowledge^{3}^{3}3Mathemata could mean both knowledge or mathematics. and who would nevertheless not think that strengths of his talents are worth trying out on difficult and yet unsolved problems. For my part, I do not deny that I too was once affected by the same weakness, and, to omit other things, I put not a small effort into squaring of a circle and in examination of works of others attempting it. I does not belong here to list methods which I had followed in this matter: I will mention only one, which perhaps somebody luckier will be able to improve. I had convinced myself about a certain conjecture, namely that certain sections of a straight line are possible, whose fragments are incommensurable to each other and to other straight lines in length and square, yet commensurable to parts of area or circumference in length or square; so that by finding the section with this method, one might procure from it a quadrature of the circle geometrically, or at least compute the ratio of the diameter and circumference with as many digits as one likes. \pend
It seems that I have arrived to this idea not blindly, but guided by a quadratrix^{4}^{4}4Quadratrix of Hippias is a curve with equation . It can be used to solve the problem of squaring the circle, although this is not a pure “ruler and compass” solution., invented by Dinostratus^{5}^{5}5Dinostratus (ca. 390 B.C. - ca. 320 B.C) was a Greek mathematician and geometer, a disciple of Plato.. And while my mind was diverted from this work by other separate pursuits, eventually, inspired by examples of great men, I turned to investigation of certain profits pertaining to cyclometry, useful in mechanical practice, as much numerically as geometrically. \pend
DIAMTERI AD PERIPHERIAM CIRCULI
Rationes Arithmeticæ^{6}^{6}6Arithmetic ratios of diameter and circumference of a circle. Ratios representing lower (“defective”) bounds are on the left, upper (“excessive”) bounds on the right.
Defectivæ | Excessivæ. | ||
A | 1. ad 3.
† |
Aa | 1. ad 4. — |
B | 8. ad 25.
† |
Bb | 7. ad 22. — |
Z | 1…. 15…. 3. | Zz | 1…. 16…. 3. |
C | 106. ad 333.
† |
Cc | 113. ad 355. — |
Y | 1…. 4697…. 3. | Yy | 1…. 4698… 3. |
D | 530762. ad 1667438
† |
Dd | 530875. ad 1667793. — |
X | 1…. 5448^{7}^{7}7A misprint in the original text, should be 5548.…. 3. | Xx | 1…. 5449^{8}^{8}8Another misprint, should be 5549.…. 3. |
E | Diam. 2945 294501. | Ee | Diam. 2945 825376. |
Periph. 9252 915567
† |
Periph. 9254 583360. — | ||
V | 1…. 14774…. 3. | Vv | 1…. 14775…. 3. |
F | Dia. 43 521624 105025. | Ff | Dia. 43 524569 930401 |
Per. 136 727214 560643
† |
Per. 136 736469 144003 — |
Harum qædam minoribus terminis,^{9}^{9}9Of these [ratios], some expressed in reduced form. Here, cc, d, and e are reduced forms of respectively Cc, D, and E, e.g. .
cc | 22 | ad | 71 — |
---|---|---|---|
d. | 265381. | ad | 833719
† |
e. | 30685681. | ad | 96401910
† |
Methodicam prædictorum Numerorum Synthesin in Cogitatis, & Inventis Polymathematicis, quæ, si DEUS vitam prorogaverit, utilitati publicæ destinavi, plenius exponam; sufficiet interim ad eorum notitiam insinuasse sequentia. Numeri Characteribus Z. Y. X. V. tam simplicibus, quam geminatis insigniti, sunt Genitores, e quorum ductu, Numeri illis subjecti C. D. E. F. simplici, geminoque charactere notati, procreantur hoc modo. Ratio 7. ad 22. Excessiva, ducta in Genitorem Z. 15; & adjecto ad Productum Diametri, numero 1. ad Peripheriæ autem, hoc altero adjacente 3; constituit Rationem C.106. ad 333, Defectivam: Genitor autem major Zz.16, ductus in eosdem terminos Excedentes 7. ad 22, adjectisque ad horum Producta numeris 1 & 3, conficit Rationem CC. 113, ad 355. Excess: \pend
Similiter hi termini Excessivi 113, 355, multiplicati per Genitores Y.Yy. videlicet 4697 & 4698. servata adjectione numerorum 1. & 3 ad Producta Diametri Peripheriæque, offerent terminos Rationum D. &Dd, quæ longe propius accedunt ad Archimedeam, a Ludolpho, & Grümbergero nostro vastissimis expressam numeris. Eadem ratione in reliquorum Terminorum genesi proceditur. Ut autem oculis ipsis usurpare liceat, quantum exactitudinis adferant Rationes illæ, visum est hoc loco adjicere Synopsin totius calculi, quo prædictæ Rationes ad Archimedeam, tanquam ad lapidem Lydium examinantur, ut appareat, quantum sit uniuscujusque peccatum, defectu vel excessu Peripheriæ taxato in partibus Diametri totius, in particulas Decimales subdivisæ. \pend
I will explain the aforementioned method more completely in Polymathic thoughts and inventions, which work, if God prolongs my life, I have decided to put out for public benefit. In the meanwhile, for acquaintance with this method, the following introduction will suffice. Numbers denoted by both single and double characters Z, Y, X, V are Originators, from which numbers subjected to them, denoted by single and double characters A, B, C, D, are derived this way. Ratio of 7 to 22, excessive, is multipied by the Originator Z. 15, and with added product of the diameter, equal to 1, and 3, close to circumference, yields the defective ratio of 106 to 333.^{10}^{10}10Fraction is transformed into . Moreover, the major originator Zz.16, multiplied by the same exceeding bounds 7 and 22, and with numbers 1 and 3 added to the products, makes excessive ratio CC, 113 to 355.^{11}^{11}11This produces . \pend
Similarly, those excessive bounds 113, 355, multiplied by originators Y and Yy, that is, 4697 and 4698, keeping addition of numbers 1 and 3, yield bounds^{12}^{12}12These bounds are and . on the ratio D and Dd, which come far closer to the Archimedean ratio, expressed by Ludolph^{13}^{13}13Ludolph van Ceulen (1540 – 1610) was a German-Dutch mathematician who calculated 35 digits of . and our Grum̈berger^{14}^{14}14Christoph Grienberger SJ (1561 – 1636) was an Austrian Jesuit astronomer, author of a catalog of fixed stars as well as optical and mathematical works. by great many digits. Remaining bounds are produced by proceeding in the same manner. In order to see accuracy of these ratios with one’s own eyes, it seemed fit to add in this place synopsis of all calculations, by which the aforementioned ratios are tested against Archimedean ratio like against the Lydian stone^{15}^{15}15Lydian stone (touchstone) – stone used to test gold for purity., so that it would become evident how big was the error of each of these ratios, with defect or excess of the circumference expressed as parts of the entire diameter, and with digits divided into small groups. \pend
Examen Rationum Cyclometricarum.^{16}^{16}16Examination of cyclometric ratios.
Diam. | 100000 | 00000 | 00000 | 00000 | 00000 | Archimedis |
Periph. | 314159 | 26535 | 89793 | 23846 | 26433 | – Ratio. |
B. | 312500 | 00000 | ||||
16519 | 26535 | Defectus^{17}^{17}17Defect, that is, the value of . Similarly, excess (lat. excessus) is the value of . | ||||
Bb. | 314285 | 71428 | ||||
126 | 44892 | Excessus. | ||||
C. | 314150 | 94339 | 62264 | |||
8 | 32196 | 27592 | Defectus | |||
Cc. | 314159 | 29203 | 53982 | |||
2667 | 64189 | Excessus | ||||
D. | 314159 | 26535 | 81077 | 77120 | ||
8715 | 46725 | Defect. | ||||
Dd. | 314159 | 26536 | 37862 | 02024 | ||
48068 | 78178 | Excess. | ||||
E. | 314159 | 26535 | 89787 | 82814 | ||
5 | 41031 | Defect. | ||||
Ee. | 314159 | 26535 | 89796 | 49172 | ||
3 | 25326 | Excess. | ||||
F. | 314159 | 26535 | 89793 | 23833 | 89913 | |
12 | 36520 | Defectus. | ||||
Ff. | 314159 | 26535 | 89793 | 23855 | 91866 | |
9 | 65432 | Excessus. |
Ex hac Tabella colligitur: imprimis quantitas Defectus, vel Excessus cujusvis Rationis, taxata Fractione, cujus Denominator est Diameter supremo loco posita, videlicet 1 cum tot zeris, quot libet assumere: Numerator autem erit is, qui in eodem cum Denominatore gradu Decimali consistit. Sic Rationis C Defectum metitur hæc Fractio quæ exactior erit, si prolixior Denominator assumatur. \pend
Colligitur ex eadem secundo. Rationes nobis exhibitas, adeo compendiosas esse, ut earum nonnullæ, duplo pluribus notis Archimedeis æquivaleant ; quanquam ipsa Cc duplum earum excedat, quæ proinde brevitate, nec non exactitudine sua, in Praxi cæteris præferenda videatur, cui, dum quid accuratius quæritur, ipsa d succedat. Præter has quidem mihi suppetunt adhuc plures, consimili dote præditæ, sed eas, ne nimius videar, alteri occasioni servandas existimo. Concludam interim singulari quadam, & ut ita dicam, curiosa Ratione, quæ est 991 ad , quæ cum Archimedea consentit in octonis notis prioribus, ac tum primum illam incipit excedere, minus quam 23 centesimis. \pend
GRAMMICÆ RATIONES CYCLOMETRICÆ,
Ad Usus Mechanicos.
Harum quidem complures olim a me repertæ; hoc tamen loco visum mihi est eam tantum proponere, quæ huic Anno præsenti, quo ista scribimus, affinitate quadam conjuncta est.
Oporteat igitur Semiperipheriæ B C D Rectam proxime æqualem reperire. Ducantur Tangentes B G, D H, quarum prior Radio AC æqualis, & jungantur GCH. Tum Radio CA secentur ex C arcus utrinque æquales CE & EF: quorum quivis complectetur Gradus 60, reliqui autem BE, DF singuli gr. 30. Agatur per E Secans AI, determinans Tangentem BI. Capiatur tandem HL, æqualis Diametro BD; ac tum ducatur IL. \pend
From this table one infers, first of all, quantities of the defect or excess of any ratio, estimated by a fraction whose denominator is the diameter placed in the initial position, with as many zeros as one wants to take. Numerator, on the other hand, is this one, which takes the same position as the denominator. Thus the defect of the ratio C is estimated by the fraction , which would be more accurate if one took a longer denominator. \pend
From the same table, a second thing is inferred. Ratios exhibited by us are so advantageous, that some of them are equivalent to Archimedean ratio with twice as many digits as others; Yet Cc itself twice exceed others^{18}^{18}18That is, it exceeds B, Bb, and C in accuracy., hence in practice by shortness and accuracy it seems to be preferred to others. If one sought a more accurate one, d would be a successor. Besides those, I have at hand indeed even more of them, similar in quality to the mentioned ones, but, in order not to appear excessive, I consider saving them for another occasion. I will, in the meanwhile, conclude with a certain singular, and so to speak curious ratio, which is 991 to 3113, which agrees with Archimedean in the first 8 digits, and then it starts to exceed it, by less than 23 hundredths.^{19}^{19}19Defining , we have . \pend
GEOMETRIC CYCLOMETRIC CONSTRUCTIONS,
For Use by Mechanics.
Of which several were once found by me. In this place, nevertheless, it seemed appropriate to present only one, associated with the current year, in which we write this.
It would be then required to find a straight line nearly equal to the semicircle BCD. Let tangent lines BG, DH be drawn, equal to the radius AC and connected by GCH. Then from C, let both parts of the arc be cut by CE and EF^{20}^{20}20This should likely be CF., equal to the radius CA. Each of them will embrace the angle of 60 degrees, while the remaining angles BE, DF will be 30 degrees each. Let a line AI be driven through E, determining the extent of the tangent BI. Finally, let HL be taken equal to the diameter BD; and then let IL be drawn.
Dico Inprimis IL æqualem esse
Semiperipheriæ BCD proxime.
Demonstratur calculo Trigonometrico. Intelligatur
autem ducta esse IK, quæ
Tangentes BI, DK conjungat.
Quoniam ad Radium
AB. 100000 00000 00000.
Tangens gr. 30 est
BI. 57735 02691 89626. Erit hujus
Compl. ad Radium, ipsa
IG. 42264 97308 10373. Igitur
Tota KH HL, sive
KL. 2 42264 97308 10373. Ergo
IK q XL q.
9 86923 17181 95572 75995 52843 99129.
Horum Radix est.
IL. 3 14153 33387 05093. Hæc autem
Deficit ab Archimedea.
Z. …. 5 93148 84700.
Continetur in AB. vicib.
X. …. …. 16859.
\pend
Dico deinde, Peripheriam sic inventam, ab Archimedea veræ proxima deficere minori Ratione, ea, quam habet Unitas ad Decuplum currentis Anni 1685 a Christo nato, Æra vulgari numerati; majore autem, quam eadem habeat ad decuplum anni 1686, proxime secuturi. Cum enim Peripheria nostra IL, ab Archimedea (præcedentis Tabulæ) deficiat numero Z, qui totam Diametrum, numero AB taxatam, metitur numero X: manifestum est, Unitatem ad numerum præsentis Anni decuplum, videlicet 16850, majorem habere rationem quam ad X, priore majorem: minorem autem, quam ad hunc 16860, per demonstrata Prop. 8 Quinti Elem. Euclid. E quibus Praxeos istius , cum intellectu comprehendi, tum memoria facile retineri poterit. \pend
Epimetri loco adjungam alteram Praxin Linearem Mechanicorum Circino opportunissimam, quod ea continua Diametri bisectione peragatur, sitque longe exactior præcedente: sic autem instituitur. Dati Circuli Diametrum, Circino bisectioni destinato, divide in partes 32. Talium enim Peripheria erit , hoc est, erit eorum Ratio, 1024. ad 3217. In Praxi igitur, Triplo Diametri, sive partibus 96, adjiciendæ erunt , sive totius Diametri, & insuper Semis unius Trigesimæ secundæ, cum alterius Semissis particula decima sexta. Hujus exactitudinem probat calculus, quo provenit Peripheria P. 314160156.– quæ Archimedeam excedit numero Q …….891, qui minor est Defectu Z, Peripheriæ præcedentis. Quanquam nec istud subticendum sit, istam Praxin in Majoribus Circulis potissimum locum habere, in parvis oculorum effugere, quoad particulam, postremo addendam. \pend \endnumbering
I say in the first place that IL is nearly equal to the semicircle BCD. This is demonstrated by trigonometric calculations. Let us assume that
the line IK is drawn, which connects tangents BI, DK.
Since, with the radius
AB. 100000 00000 00000,
tangent of 30 degrees is
BI. 57735 02691 89626.
Its complement^{21}^{21}21. to the radius, IG itself, will be
IG. 42264 97308 10373. Therefore,
together KH HL, or
KL. 2 42264 97308 10373. Hence
IK q XL^{22}^{22}22Obviously a misprint, should be KL instead of XL. IK q KL q. means . q.
9 86923 17181 95572 75995 52843 99129,
of which the root is
IL. 3 14153 33387 05093.^{23}^{23}23. But this is
short of Archimedean by
Z. …. 5 93148 84700,^{24}^{24}24.
contained in AB approx.
X. …. …. 16859.^{25}^{25}25.
\pend
I say then that the circumference found this way differs from the Archimedean ratio by less then the ratio of one to ten times the date of the current year 1685 after Christ, numbered with the common era, and more than one to ten times the date of the next year to come, 1686. When indeed our periphery IL is short of the Archimedean (of preceding table) by the number Z, which measures the whole diameter, expressed by the number AB, with the number X: it is clear that the ratio of the unity to ten times the current year, or 16850, is larger than the ratio of 1 to X, and less than 1 to 16860, as demonstrated by prop. 8 of the 5th book of Euclid^{26}^{26}26Prop. 8 of Euclid’s book 5 says: Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.[1]. From this exactness but also easiness of the construction, when embraced by the intellect, can easily be retained in memory. \pend
As a supplement, I will add another linear construction suitable for the compass of mechanics, which would be carried out by successive bisections of the diameter, and would be far more exact than the previous one: it is set up as follows. Given the diameter of the circle, intended for the bisection by compass, divide it into 32 parts. Of such kind, the circumference will be , that is, will be the ratio of 1024 and 3217. In practice, therefore, to three diameters, or 96 parts, will be added, or of the total diameter^{27}^{27}27Clearly, the author means here., and, moreover, one half of 32nd, with of another particle’s half.^{28}^{28}28This amounts to Calculations prove exactness of this procedure, yielding Circumference P. 3.14160156, which exceeds Archimedean by the number Q …….891, which is smaller than the defect Z, of the preceding circumference^{29}^{29}29.. In spite of this, one must not be silent about the fact that this construction has its place principally applied to larger circles, yet in smaller circles it is beyond one’s ability to see, especially with respect to the small particle added at the end. \pend \endnumbering
References
- [1] Euclid. The thirteen books of the Elements, volume 2. Dover, New York, 1956.
- [2] Adam Adamandy Kochański. Observationes cyclometricae ad facilitandam praxin accomodatae. Acta Eruditorum, 4:394–398, 1685.