Observationes Cyclometricae by Adam Adamandy Kochanski - Latin text with annotated English translation

📝 Abstract

Annotated parallel text in Latin and English of the classic paper of Adam Adamandy Kocha'nski “Observationes Cyclometricae”, published in Acta Eruditorum 4:394–398 (1685), in which he presented approximate geometric method of rectification of the circle, as well as a series of rational approximations of \pi. The included Latin text of Observationes closely follows the original text published in Acta Eruditorum. 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.

💡 Analysis

Annotated parallel text in Latin and English of the classic paper of Adam Adamandy Kocha'nski “Observationes Cyclometricae”, published in Acta Eruditorum 4:394–398 (1685), in which he presented approximate geometric method of rectification of the circle, as well as a series of rational approximations of \pi. The included Latin text of Observationes closely follows the original text published in Acta Eruditorum. 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.

📄 Content

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. ADAMI ADAMANDI E SOCIET. BY ADAM ADAMANY FROM THE JESU SOCIETY OF JESUS Kochanski Dobrinniaci, Sereniss. Kochański of Dobrzyń1, Mathematician and 5 Poloniarum Regis Mathematici Librarian of the Most Serene King2 of Poland, 5R & Bibliothecari, OBSERVATIONES Cyclometric OBSERVATIONS, Cyclometricæ, accommodated for easiness of practical use; ad facilitandam Praxin accomodatæ; from a letter to fellow readers of Acta. ex Epistola ad Actorum Collectores. 10 Qui Mathemata serio coluerit, nec tamen I suppose one could hardy find anyone who ad difficillima quæque & adhuc insoluta Pro- would seriously cultivate knowledge3 and who 10R blemata vires ingenii sui pertentandas cen- would nevertheless not think that strengths of his suerit, vix quenquam repertum esse existimo. talents are worth trying out on difficult and yet 15 Haud equidem diffiteor, me quoque olim eo- unsolved problems. For my part, I do not deny dem morbo laborasse, & ut alia præteream, that I too was once affected by the same weak- in Circulo quidem quadrando, vel examinan- ness, and, to omit other things, I put not a small 15R dis aliorum in eo conatibus, operæ non nihil effort into squaring of a circle and in examination collocasse. Non attinet hic enumerare Metho- of works of others attempting it. I does not be- 20 dos, quas ea in re secutus fueram: unam tan- long here to list methods which I had followed in tum, quam fortasse quispiam felicius excole- this matter: I will mention only one, which per- re poterit, commemorabo. Persuaseram mi- haps somebody luckier will be able to improve. I 20R hi conjectura quadam, possibiles esse aliquas had convinced myself about a certain conjecture, Rectarum sectiones, quarum segmenta invi- namely that certain sections of a straight line are 25 cem, & cum aliis rectis Longitudine vel Po- possible, whose fragments are incommensurable 1Dobrzyń nad Wisłą – Kochański’s birthplace, a town in Poland on the Vistula River, with settlement history dating back to 1065. 2John III Sobieski (1629 – 1696), from 1674 until his death King of Poland and Grand Duke of Lithuania. 3Mathemata could mean both knowledge or mathematics. 1 arXiv:1106.1808v1 [math.HO] 9 Jun 2011 tentia incommensurabilia essent, Circuli ta- to each other and to other straight lines in length men Areæ, vel Peripheriæ partibus Longitu- and square, yet commensurable to parts of area 25R dine aut Potentia commensurarentur; ita ut or circumference in length or square; so that by inventa sectione istiusmodi, liceret ex ea Te- finding the section with this method, one might 30 tragonismum expedire Geometrice, vel saltem procure from it a quadrature of the circle geo- rationem Diametri ad Ambitum, in numeris metrically, or at least compute the ratio of the ad lubitum maximis supputare. diameter and circumference with as many digits 30R as one likes. Ad eam porro cogitationem videbar mihi It seems that I have arrived to this idea not non temere, sed illius Quadratricis, a Dino- blindly, but guided by a quadratrix4, invented 35 strato inventæ, ductu devenisse. At cum ab by Dinostratus5. And while my mind was di- istis laboribus ad alia disparata studia animus verted from this work by other separate pursuits, 35R avocaretur, illum tandem adjeci, & quidem eventually, inspired by examples of great men, I magnorum Virorum exemplis incitatus, ad in- turned to investigation of certain profits pertain- vestiganda compendia quædam Cyclometrica, ing to cyclometry, useful in mechanical practice, 40 Praxibus mechanicis utilia, idque tam in Nu- as much numerically as geometrically. meris, quam Lineis; quorum nonnulla hoc loco adferre lubet. DIAMTERI AD PERIPHERIAM CIRCULI Rationes Arithmeticæ6 Defectivæ Excessivæ. A

1. ad 3. † Aa
2. ad 4. — B
3. ad 25. † Bb
4. ad 22. — Z 1…. 15…. 3. Zz 1…. 16…. 3. C
5. ad 333. † Cc
6. ad 355. — Y 1…. 4697…. 3. Yy 1…. 4698… 3. D
7. ad 1667438 † Dd
8. ad 1667793. — X 1…. 54487…. 3. Xx 1…. 54498…. 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 — 4Quadratrix of Hippias is a cur

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