Archimedes in the Middle Ages. III. The Fate of the Medieval Archimedes. Part iii

957 Pages·1978·28.85 MB·English

by  Marshall Clagett

Memoirs of the
AMERICAN PHILOSOPHICAL SOCIETY
held at Philadelphia
for Promoting Useful Knowledge
Volume 125 Part B
ARCHIMEDES
in the Middle Ages
VOLUME THREE
THE FATE OF THE MEDIEVAL ARCHIMEDES
1300 to 1565
Part III: The Medieval Archimedes in
the Renaissance, 1450-1565
MARSHALL CLAGETT
THE AMERICAN PHILOSOPHICAL SOCIETY
Independence Square
Philadelphia
1978
Copyright © 1978 by The American Philosophical Society
Library of Congress Catalog Card Number 76-9435
International Standard Book Number 0-87169-125-6
US ISSN 0065-9738
PART III
The Medieval Archimedes in
Renaissance, 1450-1565

CHAPTER 1
The Medieval Archimedes Toward
the Middle of the Fifteenth Century
I. Nicholas of Cusa
It is not my intention in this volume to chart the full spread and influence
of Archimedes in the Renaissance, although this is a most important sub­
ject and ought to receive more precise consideration than it has in the past.
Rather, I should like to complete my study of the role played by the
Moerbeke translations and other versions of Archimedes in the Middle
Ages by showing their continuing life in the Renaissance, trying to deter­
mine how much of the new Archimedes was still dependent on the old.
This continuing influence is not merely a fancy, for it will become evident
from our study that a good many Renaissance authors still drew some of
their Archimedean knowledge from the medieval traditions and that in
fact the first published complete Archimedean texts were those of the .
Moerbeke translation.
I have already discussed such authors as Leonardo de Antoniis of
Cremona and Giovanni Fontana who belong to the first half of the
fifteenth century exclusively and who seem to belong wholly to the
medieval traditions. It is now my intention to investigate authors writing
toward the middle of the century. The first of these is the celebrated
Nicholas of Cusa. Cusa’s treatment of Archimedes, never extensive,
can be divided into two chronological periods. The first lies before 1453
and includes those works which show some knowledge of the medieval
Archimedes but no precise knowledge of the actual Archimedean
texts. The second—following 1453—reveals some knowledge of the
actual texts. We know that just before the composition of his De mathe-
maticis complementis (Book I completed in 1453 and Book II in 1454)
Nicholas of Cusa received from Pope Nicholas V the text of the newly
translated corpus of Archimedean works, for Cusa dedicates his work to
the Pope and, among other things, tells us:1
1 Nicholaus Cusanus, Opera, Vol. 2 (Paris, 1514), 59r (cf. the edition of P. Wilpert
drawn from the edition of Strasbourg of 1477, Nikolaus von Kues, Werkc, Vol. II [Berlin,
1967], p. 388): "Tradidisti enim mihi proximis diebus magni Archimedis geometrica.
297
298 ARCHIMEDES IN THE MIDDLE AGES
For you have in recent days transmitted to me the geometrical works of the great
Archimedes, presented to you in Greek and converted into Latin by your
endeavor. They appeared to me so remarkable that I was able to become versed
in their subject only with great diligence; and the result of this was that I was able
by my endeavor and labor to add something to them and I decided to offer it to
Your Holiness.
This translation was, without doubt, the translation ordered by Nicholas
' V and executed by Jacobus Cremonensis (see below, Chapter 2, Section
I). We see a number of traces of the influence of this translation on
Nicholas of Cusa in the preparation of theDe mathematicis complementis.
For example, in one place2 we are told that Archimedes has proved that
the surface of a sphere is equal to four of its great circles, which Nicholas
had presumably read in Archimedes’ On the Sphere and the Cylinder,
Book I, Proposition 33 (Prop. 31 in the Cremonensis translation). Nicholas
also tells us later in the same work3 that “from Archimedes you have the
art of reducing every segment of a sphere to a plane circle,” an obvious
reference to Book I, Propositions 42-43 (Props. 40-41 in the Cremonensis
translations) of the same work. Further, we are told a few lines later,
“. . . Archimedes in the Quadrature of the Parabola has already made
clear how that surface can be reduced to a square, showing that the surface
bounded by a straight line and the section of a right-angle cone is four-thirds
the triangle having as its base the straight line of the parabola and as its
altitude [that] of the parabola.”4 Needless to say, the source of this state­
ment was Proposition 17 of On the Quadrature of the Parabola, no doubt
as rendered in the new translation rather than in that of Moerbeke.
Nicholas also speaks in the same work of the use of a spiral line for the
rectification of a curved line:5
graece tibi praesentata et tuo studio in latinum conversa, quae mihi tam admiranda visa
sunt: ut cura (/ circa) ipsa non nisi magna cum diligentia versari potuerim; ex quo id effectu m est:
ut meo studio et labore complementum aliquod illis addiderim, quod tuae sanctitati offerre
decrevi.” (Punctuation in this and succeeding passages from Cusa’s works has been slightly
altered.) Cf. also Schriften des Nikolaus von Cues im Auftrage der Heidelberger Akademie
der Wissenschaften in deutscher Ubersetzung, ed. of E. Hoffmann, Vol. 11: Die mathe-
matischen Schriften, tr. of Josepha Hofmann with introduction and notes by J. E.
Hofmann (Hamburg, 1952), pp. 68-69. I have followed this volume for the dates of the
various mathematical writings of Nicholas of Cusa. Note that all of my page references
to Volume 2 of the 1514 edition of Cusanus’ works are to the second set of page numbers
in that volume.
2 Opera, Vol. 2, 72r: ". . . et ab maximus circulus, cuius quatuor superficies aequantur
superficiei spherae, ut probat Archimedes.”
3 Ibid., 77v: "Et quoniam ex Archimede artem habes, omnem portionem superficiei
spherae in circularem planam reducendi. . . .”
iIbid., “Ex quo iam patefecit Archimedes in quadratura parabolae: quomodo super­
ficies ilia potest in quadratam reduci, ostendens superficiem illam ex recta et sectione
coni rectanguli esse sesquitertiam ad triangulum habentem basim ipsam rectam parabolae
et altitudinem ipsius parabolae.” (Note: 1 have capitalized Archimedes in the first line.)
5 Ibid., 59r; “Testimonio omnium qui se ad geometrica contulerunt: nemo propinquius
Archimede ad circuli pervenit quadraturam, qui videns illam attingi non posse nisi curva
NICHOLAS OF CUSA AND ARCHIMEDES 299
On the testimony of all who have concerned themselves with geometry no one
has come closer to the quadrature of the circle than Archimedes, who, seeing that
it could only be done if a curved circular line were converted into a straight line,
attempted to demonstrate this procedure by means of a spiral. But because the
ratio of the motion of a point from the center through the radius to the motion
in which another point is moved in the same time through the circumference
(without which the spiral cannot be described) is the same as the ratio of the
radius to the circumference, which is not known but which is sought, hence it
seems that he has failed [in his objective]. For it will be easier to square the
circle than to describe a spiral and to apply a tangent to it at the end of its revolu­
tion. Therefore, this art [of rectification], on the basis of what Archimedes has left
[to us], remains up to now completely obscure.
It might be thought at first glance that Nicholas composed this critique
after consulting On Spiral Lines in the translation given to him by
Nicholas V. However, as we shall indicate below, this same criticism of
the Archimedean procedure was already given by Nicholas in his
Quadratura circuit, composed in 1450 and thus was probably written
before he had examined the new translation. Hence, it appears that we
should look elsewhere for his knowledge of On Spiral Lines as represented
by the criticism presented in the De mathematicis complementis.
The first evidence of Archimedean knowledge on the part of Nicholas
of Cusa appears in the De geometricis transmutationibus of 1445, although <>
it is worth noting that Archimedes is not mentioned by name in that tract.
The initial passage of interest is that in which the so-called Platonic
solution of the problem of finding two proportional means between two
given lines is presented:8
circularis linea in rectam resolvatur, nisus est hanc artem mediante helica ostendere. Sed
quia proportio motus signi a centre per semidiametrum ad motum in quo in eodem tempore
aliud signum per circunferentiam movetur, sine qua helica describi nequit, se habet ut semi­
diameter ad circunferentiam quae non est scita sed quaeritur, hinc videtur ipsum defecisse.
Facilius enim erit circulum quadrare quam helicam describere et contingentem eidem in fine
circulationis applicare. Remanet igitur es iis quae Archimedes reliquit haec ars adhuc
penitus abscondita.” (Note: I have capitalized Remanet.) (Cf. Wilpert edition cited in note
1, pp. 388-89.)
6 Ibid., 42v: “Tertium praemissum. Tertium quod ante mittendum asservi est: quomodo
inter duas lineas rectas dime mediae continue proportionates statuantur. Iamdudum notis-
simum fuit, si datae dime lineae simul iunctae diametri circuli fiant et eas chorda ortho-
gonaliter separavit, quod semichorda est medio loco inter ipsas proportionalis, quoniam
semichorda inter sagittam et residuum diametri mediare necessarium est. Si igitur duae lineae
indefinitae longitudinis, ut ab et cd, se orthogonaiiter secuerint in e puncto, et de e versus
d minorem lineam signavero quaesit ef, et de e versus a maiorem quae sit eg, descrip-
seroque duos semicirculos: unum super centro in linea ec puta k, alium super centre in
linea ea puta h existente, hac quadam advertentia quod arcus semicirculi cuius centrum
reperitur in ea linea concurrat cum arcu alterius semicirculi in linea eb et linea ec puta
punctis i et l, nemo haesitare potest cd et el mediare, ex praemissa notissima regula unici
medii proportionalis, inter ef et eg. Unde ut in praxi lmec media facile attingas, habeto
gnomonem atque lineam unam quae ad latus gnomonis applicata rectum angulum efficiat.
Et iuxta praemissa duas indefinitae quantilatis lineas fac invicem orthogonaiiter secure.
Ponas deindc rectum angulum gnomonis super lineam eb et latus unum super/punctum,
300 ARCHIMEDES IN THE MIDDLE AGES
The Third Premise. A third thing which I have previously asserted as necessary
to premise is this: How to insert between two straight lines two continually pro­
portional means. It was already well known that, if two given lines which are
joined together become the diameter of a circle, and a chord orthogonally drawn
divides them, then the semichord is the mean proportional between them since
the semichord necessarily is a mean between the versine and the rest of the
diameter. Therefore, if two lines of indefinite length, such as ab and cd [see Fig.
III. 1.1], intersect orthogonally in point e, and from e in the direction of d I mark
off a shorter line ef, and from e in the direction of a [I mark off] a greater line eg,
and I describe two semicircles: one on center A in line ec and the other on center h
in line ea, with this warning that the arc of the semicircle which is found on line ea
intersects the arc of the other semicircle in points l and /, respectively, of lines
eb and ec, no one can hesitate [to say] that ei and el are the means between ef and
eg, following the well-known premised rule of one mean proportional. Hence,
in order to find these means easily in a practical way, take a gnomon [i.e.,
carpenter’s square] and a straight line which when applied to the side of the
gnomon produces a right angle. And, according to the things premised, cause
two lines of indefinite length to intersect orthogonally. Then you place the right
angle of the gnomon on line eb and one side on point/ and note where the other
side cuts line el. Apply the rule [i.e., the line] of which I spoke to the side at
that point so that it produces a right angle. If this rule passes through point g
you have what is sought. If not, move the gnomon up or down on line eb until
the rule does [pass through point g], and you have the two means which you seek.
• Several other methods can be easily found by one who wishes to exert effort. But
this method, since it is clear, may suffice for the present.
It is immediately evident that the ultimate source of this passage is
Eutocius’ Commentary on the Sphere and Cylinder of Archimedes, Book
II, Proposition 1. This could have been read by Nicholas directly in the
Vatican manuscript of the Moerbeke translations, or, which is more likely,
, in Johannes de Muris’ De arte mensurandi (see above, Part I, Chapter 3).
Another possible source is the treatment of the Platonic solution in the
Verba filiorum, Proposition XVII, of the Banu Musa (Vol. 1, p. 340). But
since the use of the gnomon is not so apparent in the treatment of the
Banu Musa as it is in the Moerbeke translation or its adaptation in the
De arte mensurandi, we can probably rule out the Verba filiorum as the
source of Nicholas’ discussion. The final reference by Nicholas to ‘ ‘ several
other methods” seems to confirm the conclusion that the Verba filiorum
alone was not the source of Nicholas’ description of the Platonic solution
of the problem since only one other solution is given there, while two other
methods are mentioned in the De arte mensurandi and eleven others are
et nota ubi reliquum latus secuerit lineam ec. Applica ibi regulam ad latus de quo dixi, ut
rectum angulum efficiat. Si haec regula per# transient, habes quaesitum. Si non, gnomonem
in eb attrahe, vel elonga, quousque ita evenerit, et habes ilia duo media quae inquiris.
Possunt quidem et alii plerique modi de facili inveniri per eum qui studium adhibere voluerit.
Sed hie modus, cum clarus sit, ad praesens sufficiat.” (In addition to altering the punc­
tuation and occasionally capitalizing a word, I have also italicized the letters marking the
lines.)
NICHOLAS OF CUSA AND ARCHIMEDES 301
given in Eutocius’ Commentary. It is of course possible that Nicholas
knew both the Verba filiorum and the De arte mensurandi, which together
would justify his statement that several methods can be found. As between
the De arte mensurandi and the direct Moerbeke translation, I prefer the
De arte mensurandi as Nicholas’ source for two reasons. In the first e
place, the De arte mensurandi (or at least the hybrid quadrature tract
included in the eighth chapter of the De arte mensurandi) was in all
probability the source of Nicholas’ rather inadequate knowledge of the
use of a spiral line in the rectification of a curved line, as I shall argue below,
and hence the work seems to have been known to Nicholas (and, of o
course, the existence of several Italian manuscripts of the De arte
mensurandi is an indication that it was known in Italy). In the second
place, Nicholas reveals, at this time, almost no knowledge of the other
Archimedean tracts included in the Moerbeke corpus.7 Of course, even if
Nicholas did draw his knowledge of the proportional means problem
from the De arte mensurandi, it could have been from a fragmentary
version of the pertinent proposition like that of MS Bern, Stadtbibl. A.50,
176v-77r (see above, Introduction to Part I, Chapter 3, note 7).
A further passage from the De geometricis transmutationibus gives the
basic conclusion of Proposition 1 of the De mensura circuli of Archi­
medes but without reference to Archimedes:8
7 One could put forth the argument that Nicholas saw the Moerbeke codex when it was
in the hands of his friend Paolo Toscanelli. But then one must first establish that Toscanelli
did possess the Vatican manuscript of Moerbeke’s translations. Heiberg has argued this
in the affirmative on the basis of a remark concerning the end of Book 1 of On the Equi­
librium of PUmes added by Regiomontanus to his copy of Jacobus Cremonensis' transla­
tion: "male stat. vide exemplar utrumque Domini Niceni grecum et latinum. vide etiam
exemplar vetus apud magistrum Paulum.” (MS Nuremberg, Stadtbibl. Cent. V.15, p. 139;
cf. Heiberg, Archimedis opera, Vol. 3, p. LXXI.) Heiberg reasoned (correctly, I believe)
that the expression "exemplar vetus" could only refer either to Greek codex A or to
the Vatican manuscript of the Moerbeke translations. But Heiberg’s further argument is
weaker, when he argues against the first possibility solely on the basis of the probability
that Cosirno di Medici, who knew Toscanelli well, would have taken possession of such
a valuable old codex if it was a Greek manuscript, but there is no evidence that Cosimo’s
library ever contained an old codex of Archimedes. Hence, according to Heiberg, it was
the Moerbeke codex that Paolo possessed. But even if one accepts this doubtful reasoning
and concludes that Toscanelli did have the Moerbeke manuscript in his possession (and
I shall give further arguments below for this possibility) it seems unlikely that Cusanus had
examined it as early as 1445, for, in the passage quoted above from the De mathematieis
complementis, composed in 1453, he expresses his wonder and admiration for the geometrical
tracts of Archimedes in the new translation sent to him recently by Pope Nicholas V.
If he had already examined the Moerbeke codex closely enough to make extracts from
Eutocius’ Commentary on the Sphere and the Cylinder and from Archimedes’ On Spiral
Lines, he could hardly have reacted in this fashion, for the Moerbeke codex contained
almost as many works as did the new translation. Some would perhaps say that, by the
time he wrote his Idiota de staticis experimentis (1450) he had seen the Moerbeke codex,
for in the De staticis he was clearly cognizant of the Principle of Archimedes. But I have
argued against this opinion below.
8 Ibid., 45r: "Superficiem circularem si in rectilinealem transmutare proponis, primo eius
peripheriam curvam in rectam resolvito. Deinde semidiametrum peripheriae ad rectum