1
THE
WRITINGS
OF ANTOINE PARENT
VOLUME 1: MATHEMATICS
John F. Hall
Professor of Civil Engineering, Emeritus
Caltech
Caltech Civil Engineering Report No. 2024
-
0
3
July 2024
2
T
able
of
contents
Page
Chapter
1
.
Introduction
4
Chapter
2
. Analytic geometry
1
3
Chapter 3. Projections used for making maps
2
2
Chapter
4
.
C
riti
que
of Louis Carré’s book on integra
tion
38
Chapter 5. Shape of least resistance
4
6
Chapter
6
. The hanging chain
5
4
Chapter
7
. Geometry of the honeycomb
7
7
Chapter
8
. Polyhedra
87
Chapter
9
. Supplement to the geometry of Archimedes
1
05
Chapter 10. Additional
geometry
1
4
3
Chapter
1
1
. New
p
roperty of a
c
ycloid
1
5
8
Chapter
1
2
. Grinding a hyperbolic lens
1
6
6
Chapter
1
3
. Clock fusee
1
72
Chapter
1
4
. Two papers on navigation
1
82
Chapter
1
5
. Two papers on orbiting bodies
19
5
3
Chapter
1
6
.
T
wilight
of shortest duration
2
0
4
Chapter 1
7
. Hanging chandelier
2
13
Chapter 1
8
. Arithmetic
book
2
20
Chapter 1
9
.
Casting out nines
and elevens
2
30
Chapter
20
. Optimization for engineering problems
2
3
6
Chapter
21
.
Two papers on graphical construction of curves
24
7
Chapter
2
2
.
Summary
2
62
The drawing on the front
page shows a
sphere and its circumscribed cylinder (on
the left) with a corresponding slice taken out of each. Archimedes found that the
surface areas of the slices are equal, regardless of the location and thickness of the
slice
s
,
as long as the locations and thicknesses are
the same
.
On the right is shown
an oblate spheroid and a certain circumscribed one
-
sheeted hyperboloid with a
corresponding slice taken out of each. Parent found a similar equality of the
surface areas of the slices when the asymptotic slope of
the hyperboloid is a certain
value that depends on the dimensions of the spheroid.
See subsection 2.6 of
Chapter 9 for a proof.
4
Chapter 1. Introduction
The author of this report “discovered” Antoine Parent while searching for early
papers about how to incorporate the effects of friction in the analysis of machines.
Parent’s work in this area, which occurred only a few years after the laws of
friction wer
e postulated, was not only decades ahead of other researchers, but it
demonstrates a fundamental understanding of mechanics unique for that time.
Thus began an effort by the author to learn what else Parent had accomplished, and
the first result of that e
ffort is this report on Parent’s writings in mathematics.
Future reports will cover Parent’s more expansive work in engineering
, as well as
other topics.
Section 1
. Information about Parent
Antoine Parent was a French
engineer and
mathematician, born in 1666 and died in
1716 at the age of 50. According to the
éloge
(1) read to the
Académie Royale des
Sciences
by its secretary,
Bernard Fontenelle, Parent obtained a degree in law in
Paris following the wishes of his family. However, he never practiced, instead
devoting himself to self
-
study in mathematics, science and engineering. He
attended lectures at the
Collège Royal
where he impressed J
o
seph
S
auveur with his
abilities.
It may have been Sauveur (2) who arranged for Parent to b
e admitted to
the
Académie
in 1699 as a
student member
under the sponsorship of
Gilles
Filleau
des Billettes,
who was admitted the same year.
Both des Billettes and Parent had
the title of mechanician, although des Billettes, being older, was awarded the rank
of pensioner.
Parent never seems to have obtained a formal position at a university, and so the
source of his support is unclear. He tutor
ed
pupils so this may have provided some
revenue. In any case, he lived frugally, and he never married.
5
At the
Académie
, Parent was very active and gave numerous talks on the results of
his investigations. He came to be known as someone who was interested and
knowledgeable in every topic that arose, and no doubt he was inspired by the
things that he heard, often following
up with further studies of his own. However,
Parent earned a reputation as being overly critical of others and often too hasty in
his judgements (1). As a result, he was not well regarded by some, and this may
explain why he was only able to publish 11
of his papers in the
Historie de
l’Académie Royale des Sciences
. In fact, Parent was never promoted at the
Académie
, although shortly before he died the rank of student member was
abolish
ed
in favor of adjoint, and Parent became adjoint, still the lowest rank.
Parent found other outlets to present his work, and he was quite prolific: 5 books
and on the order of 190 papers, the first
paper
appearing in 1699. The books are as
follows: a commentary and criticism of the philosophy
of
René
Descartes
published in 1705 as Part 2 of Volume 1 of
Recherches de mathématique et de
physique
,
252 pages, reference 3), a treatise on water pumps (approved by the
Académie
in 1700, but it apparently never became available), a treatise on
mechanics (published in 1700, 449 pages, reference 4), a textbook on arithmetic
(published in 1714, 236 pages, reference 5) and a textbook on geometry (published
in 1718 after Parent’s death
, 422 pages, reference 6).
The papers span a wide variety of subjects, including mathematics (analytic
geometry, applications of calculus, geometric analysis) and mechanics
(mechanical, civil, bio, wind and naval engineering) plus astronomy, navigation,
geology, oceans, physics, opt
ics, fortifications, botany, anatomy, language, music
and aesthetics. Many of the papers appear in additional parts and
volumes of
Recherches de mathématique et de physique
that were also published
in 1705 (
7
),
and
then this set of papers was expanded and republished in 1713 in three volumes
as
Essais et recherches de mathématique et de physique
(8).
Altogether, about
11
4
papers
a
re presented this way. In addition to these and the 11 papers published in
Historie de l’Académie Royale des Sciences
(1704 to 1712),
Parent ha
s
14 in the
Journal des savants
(1699 to 1701), 18 in the
Mercure galant
(1712 to 1713), and
6
11 in the
Journal de Trévoux
(1712 to 1716).
Still more were
just
inserted in the
backs of his arithmetic and geometry books, 6 in the former and 13 in the latter.
Finally, there are a few works transcribed into the proceedings of the meetings of
the
Académie,
known as the
Procès
-
verbaux
,
that
were
never
formally published
elsewhere.
Unlike others at the time, Parent did not disseminate ideas and results via letters to
colleagues. After he died, his writings, including additional work such as a proof
of the divinity of Jesus Christ in four parts, were left with the captain of the guar
ds
of the
Académie
as executer (1).
With a few notable exceptions, such as his advancements in analytic geometry and
optimization of machines, Parent’s work seems to have had very limited impact at
the time, and he is not well known today. The Wikipedia page for Parent (9)
briefly mentions
some results in analytic geometry and beam theory, in nine lines
of text. The MacTutor index (10), which contains biographies of over 3000
notable mathematicians, including 364 who were born in France, does not include
Parent. However, over the last deca
des
a few
publications have appeared that
discuss a few of Parent’s papers in some depth (11,12,13,14).
One of the factors that hampered appreciation of Parent’s work, which is entirely
written in French, is his rather difficult style of writing. Regarding Parent’s
Essais
et recherches de mathématique et de physique
of 1713,
C
lifford
Truesdell (1
1
)
describes it as
“miserably printed, confusingly paginated, and full of misprints or
errors of inadvertence; lists of corrigenda occur here and there.” Indeed, Volume 2
contains 120 pages of corrections and additions that go with 660 pages of papers.
For Volume 3
, there are 78 pages of corrections and additions and 528 pages of
papers. The papers are arranged one after another in an order that appears to be
completely random.
As a sample, two pages of text and a plate of figures are shown in Figures 1 and 2
from Parent’s paper on supplementing the geometry of Archimedes (
reference
15
,
7
74 pages long with 32 drawings in three plates
). Reading such a paper and
grasping the detail evident in the drawings is tedious and time consuming. And
compared to some others, this paper was actually fairly clearly written.
One question that arises is how Parent was able to produce so much written
material, not only from a mental capacity standpoint, but physically. He was only
active for about 17 years and was a lone individual with limited resources.
Members of the
Académie
were expected to
conduct studies and publish their
results, so maybe some assistance was provided there. In addition to the text, there
were hundreds of figures that needed to be drafted.
Specifically,
about mathematics, the historian J. Morton Briggs (14) says that
“Parent aimed at extending the power of both geometry and the new calculus,
although he was far from moving to pure analysis.”
Briggs
adds that “if Parent had
a particular characteristic, it is perhaps his sense of the practical
,” and he mentions
Parent’s “conviction that the seemingly esoteric nature of mathematics had a very
real utility.”
Section 2
. About this report
This report documents Parent’s writings on mathematics, drawing from
about 40
of
his papers plus some material from his treatise on mechanics (4) and his arithmetic
textbook (5). Selections have been made to demonstrate Parent’s broad facility
with mathematics as well as his innovations and a few practical applications.
Original
notation has been kept to the extent possible, although some changes were
necessary to improve clarity and consistency. There are
20
technical chapters in
the report, each one de
aling with a single topic or two related topics.
Every chapter includes relevant background information so that Parent’s work can
be viewed in perspective at the time of his writing. All references mentioned by
8
Parent are discussed. On the other hand,
little
attempt is made to document how
Parent’s results were subsequently used by others.
Many figures contained in this report are either redrawn from Parent’s figures or
are newly drawn. For these, 3
-
dimensional geometries are shown using only
parallel projection.
Regarding notation, the symbol
휋
was not yet in use in Parent’s time. Instead, it
was represented as a ratio like
푐
2
푟
, where
푐
is the circumference of a circle and
푟
is
the radius. For convenience,
휋
is used here. The line segment connecting points
퐴
and
퐵
is denoted by
퐴퐵
and its length by
퐴퐵
̅
̅
̅
̅
.
The symbol Log denotes logarithm
to base 10.
Section 3
.
References
1.
Bernard Fontenelle, “
Eloge de M. Parent
,
”
pages 88 to 93 of
Historie
section,
Historie de l’
A
cadémie
R
oyale des
S
ciences
, Paris, 1716.
2. David Sturdy,
Science and Social Status, The Members of the
Académie des
Sciences
1666
-
1750
,
Boydell & Brewer
, Suffolk, UK, 1995.
3.
Antoine Parent
, “
L
’
a
nalyse des trois der
nière
s parties des principes de la
p
hilosophie de M. Descartes
,
” pages 1 to 252,
Recherches de
m
athématique et de
p
hysique
, Part 2 of Volume 1,
Chez Jean Jombert et Florentin Delaulne
,
Paris,
1705.
4.
Antoine Parent,
Elemens de
mécanique et de physique
,
Chez Florentin & Pierre
Delaulne
, Paris, 1700.
5. Antoine Parent,
Traité d'arithmétique théori
-
pratique
en sa plus grande
perfection
,
Chez Jean de Nully
et Claude Jombert
,
Paris,
1714
.
9
6.
Antoine Parent,
Suite des
éléments de mathématique, contenant la géométrie
théorie
-
pratique, démontrée dans un ordre nouveau, & par des preuves nouvelles,
simples & généraux, dont les figures sont tirées d’Euclide, Archimède, & autres
auteurs qui en ont écrit depuis
,
Laurent d
’
Houry, Florentine Delaulne, Charles
Moette
,
Paris, 1718.
7. Antoine Parent,
Recherches de
m
athématique et de
p
hysique
,
Part
s 1 and
3 of
Volume 1, V
olume
2
,
Chez Jean Jombert et Florentin Delaulne
, Paris, 1705.
8. Antoine Parent,
Essais et
r
echerches de
m
athématique et de
p
hysique
,
Volume
1, V
olume
2
, Volume 3,
Chez Jean de Nully
, Paris, 1713.
9.
“Antoine Parent,”
Wikipedia
,
https://en.wikipedia.org/wiki/Antoine_Parent
10.
MacTutor Index
, School of Mathematics and Statistics, University of St.
Andrews, Scotland,
https://mathshistory.st
-
andrews.ac.uk/Biographies/
11. Clifford Truesdell,
The Rational Mechanics of Flexible or Elastic Bodies 1638
-
1788
,
Springer Science & Business Media, 1980
.
12. M. Yannick Fonteneau
, Chapter 2
, “
P
uissance
continuelle
et
effet
général
:
antécédents
du
travail
mécanique
dans
les
œuvres d
’A
montons
et
de
P
arent
(1699
-
1714)
,
”
Développements précoces du concept de travail mécanique (fin 17
e
s.
-
début 18
e
s.) : quantification, optimisation et profit de l'effet des agents
producteurs
,
Thesis, University of Lyon
, 2011.
13. Jed Buchwald, “Discrepant Measurements and Experimental Knowledge in
the Early Modern Era,” pages 565 to 649,
Archive for History of Exact Sciences
,
Volume 3, Springer, 2006.
10
14. J. Morton Briggs, Jr., “Parent, Antoine”,
Encyclopedia.com
,
https://www.encyclopedia.com/science/dictionaries
-
thesauruses
-
pictures
-
and
-
press
-
releases/parent
-
antoine
15.
Antoine Parent,
Memoir
XXXI, “
Supplément sur la
g
éométrie, & sur la
s
tatique d’Archimède
”, pages 453
-
526,
Essais et recherches de mathématique et de
physique
, Volume 3,
Chez Jean de Nully
, Paris, 1713.
11
Section 4
. Figures
Figure 1.
Two pages from Parent’s paper on supplementing the geometry of
Archimedes (15).
12
Figure 2. A plate of drawings from Parent’s paper on supplementing the geometry
of Archimedes (15).