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Lntered according to Aci of Congress mitu year 1833 bjH I JionJitcti in the Cleii's Oflice of Ihe bistncl Court of Massadiusetts
MjfcCANIQUE CELESTE.
BY THE
MARQUIS DE LA PLACE,
PEER or riUNCi ; crand cross or thk leoion or honor; member or the erench academy, or the academy
or sciEXcEs or paris, or the board or longitude or rRANCE, or the royal societies or
LONDON AND GOTTINGCN, Or THE ACADEMIES OE SCIENCES OF RUSSIA, DENMARK,
SWEDEN, PRUSSIA, HOLLAND, AND ITALY; MEMBER OF TUB
AMERICAN ACADEMY OF ARTS AND SCIENCES ; ETC.
TRANSLATED, WITH A COMMENTARY,
BT
NATHANIEL BOWDITCH, LL. D.
YELLOW or THE ROYAL SOCIETIES OE LONDON, EDINBURGH, AND DUBLIN ; OE THE ASTRONOMICAL SOCIETY
or LONDON, or THE PHILOSOPHICAL SOCIETY HELD AT PHILADELPHIA, OE THE AMERICAN
ACADEMY or ARTS AND SCIENCES ; CORRESPONDING MEMBER OF THE ROYAL
SOCIETIES OF BERLIN, PALERMO ; ETC.
* VOLUME IV.
WITH A MEMOIR OF THE TRANSLATOR,
BY HIS SON,
NATHANIEL INGERSOLL BOWDITCH.
BOSTON:
PROM THE PRESS OF ISAAC R. BUTTS ;
CHARLES C. LITTLE AND JAMES BROWN, PUBLISHERS.
M DCCC XXXIX.
Entered, according to Act of Congress, m the year 1839, by
N. I. BowDiTCH and J. Ingersoll Bowditch,
in the Clerk's Office of the Disuict Court of Massachusetts.
MEMOIR.
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THIS
TRANSLATION AND COMMENTARY
ARE DEDICATED, BY THE AUTHOR,
c:o ttr iWtmorg of ^in 223(fe,
MARY BOWDITCH;
WHO DEVOTED HERSELF TO HER DOMESTIC AVOCATIONS WITH GREAT JUDGMENT, UNCEASING
KINDNESS, AND A ZEAL WHICH COULD NOT BE SURPASSED ;
TAKING UPON HERSELF THE WHOLE CARE OF HER FAMILY,
AND THUS PROCURING FOR HIM THE LEISURE HOURS TO PREPARE THE WORK ;
AND SECURING TO HIM,
BY HER PRUDENT MANAGEMENT,
THE MEANS FOR ITS PUBLICATION IN ITS PRESENT FORM,
WHICH SHE FULLY APPROVED ;
AMD
WITHOUT HER APPROBATION THE WORK WOULD NOT HAVE BEEN UNDERTAKEN.
MEMOIR.
The children of the late Dr. Bowditch feel assured that every
reader of this Translation and Commentary will be desirous to
know something of the life and character of him who planned
and executed so vast a work, and of her to whose memory it is
dedicated. We have been blessed in our parents far beyond the
lot of others. Such a father and such a mother are but rarely
given by Heaven to any one. Both now sleep in the grave ; and
our kindred dust will soon be mingled with theirs in that last
resting-place. But after the lapse of many years, this work,
devoted to the elucidation of one of the most abstruse and profound
subjects of human investigation, will still endure, a memorial of
the genius of its author. Upon this monument we would inscribe
our filial tribute — a record of the parents' virtues, of the children's
gratitude and affection.
Nathaniel Bowditch was born at Salem, in Massachusetts,
March 26, 1773, being the fourth of seven children of Habakkuk
Bowditch, by his wife Mary, who was the daughter of Nathaniel
Ingersoll. His ancestors had always resided in that place from
its earliest settlement, having been, for the four last generations,
ship-masters. Tradition has handed down the fact, that three
VOL. IV. c
10 MEMOIR.
brothers, Jonathan, Joel, and WiUiam Bowditch, emigrated to this
country from England, and, as is believed, from the city of Exeter,
or its immediate vicinity. William became an inhabitant of Salem
in 1639. His humble situation in life may be inferred from the
title of " Goodman," by which he is mentioned, as distinguished
from the more dignified appellation of " Mr." He was living in
1649, in which year he had a grant of thirty acres of land from
the town. The time of his death is unknown.
He left an only child, of the same name, who was born in
1640, and died in 1681. He was collector of the port of Salem
under the Colonial government, and owned a warehouse and
other real estate, and several small vessels, but died insolvent.
He likewise left an only child, also named William, who was born
in September, 1663, and died May 28, 1728, in the sixty-fifth
year of his age. He was actively engaged as a ship-master for
many years, and was well known as an enterprising merchant. A
dangerous rock in the channel of the harbor, of which the original
Indian name was *' Tenapoo," still bears the name of " Bowditch's
Ledge," which was given to it in consequence of a vessel called
the ** Essex Galley," under his command, having struck upon it
about the year 1700. He was for many years one of the selectmen
for managing the affairs of his native town, and served also, during
several sessions, as a representative in the General Court of the
Province. He married, August 30, 1688, Mary, the daughter of
Thomas Gardner, Esq., a wealthy merchant. She died in 1724,
four years before her husband. He left an estate valued at between
four and five thousand pounds. The grave-stones of both husband
and wife are still to be seen in the burial-ground at Salem,
MEMOIR. 11
though the inscriptions are partially effaced. There were eleven
children from this marriage, the seventh of whom was Ebenezer,
who was born April 26, 1703, and died February 2, 1768, also
in the sixty-fifth year of his age. He was the only son of this
numerous household who left male descendants, and thus became
the common ancestor of all who now bear the family name in
Salem. He married, August 15, 1728, Mary, the daughter of
the Hon. John Turner, one of the most distinguished citizens of
Salem, long a member of the Provincial Council, and well known
in the local history of that time. The annalist of Salem says of
him, "His deserts were equal to his honors."*
Ebenezer Bowditch preserved through life an irreproachable
character, and possessed in a high degree the confidence and
respect of the community. He pursued his father's occupation,
but, as it seems, without much success, since he left but little
property at his death. His wife survived him till May 1, 1785,
living in reduced circumstances, and being dependent upon her
young grandson, the subject of this memoir, for many little
attentions, by which her declining years were rendered more
comfortable. They had six children, of whom the fifth was
Habakkuk, born in 1738. He also was in early life a ship-master,
and, as was the custom of that day, learned the trade of a
cooper, a practical acquaintance with which was then deemed
an important, though subordinate qualification for the discharge
of the appropriate duties of his situation. At the commencement
of the revolutionary war, he met with misfortunes in business,
* Annals of Salem, from its first Settlement ; by Joseph B. Felt ; p. 423.
\% MEMOIR.
by which his circumstances became very much reduced ; and he
was so disheartened, that he had not the energy to attempt to
retrieve them. He subsequently worked at the trade of a cooper,
which he had originally learned from the motive above stated.
But he was able in this manner to earn only a scanty subsistence
for his wife and children. Some idea may be formed of his
poverty, from the circumstance that, for many successive years,
he received fifteen or twenty dollars annually from the charity
funds of the Salem Marine Society, of which he was a member,
deriving from this sum, small as it was, quite an essential aid
toward the support of his family. He was a man very remarkable
for strong natural good sense, but had enjoyed no advantages
of education. He was an attentive observer and an intelligent
judge of men and events as they passed before him. He was
extremely conversant with the Scriptures, and entertained liberal
and enlightened views on the subjects of religion and revelation.
He possessed a cheerful temper and an amiable disposition. But
having yielded to the reverses which he had encountered, and
which would but have stimulated others to greater efforts, he
outlived those prospects of usefulness and happiness, which,
upon his entrance into life, seemed to be within his reach. Upon
one occasion, his son, alluding to the latter years of his father's
life as having been less happy than his earlier ones, expressed
the hope that he might not himself '' live too long." Habakkuk
Bowditch was married July 23, 1765, and died July 18th, 1798,
at the age of sixty years.
Mary, the wife of Habakkuk Bowditch, had died December 16,
1783. She was an excellent woman, discharging all the duties
MEMOIR. 13
of life with exemplary fidelity. By her death Dr. Bowditch
was deprived of his earliest and best friend, at the age of ten
years. But there can be no doubt that she had lived long enough
to exercise a most salutary influence over her son's mind, and
that it is in a great degree to the teachings and example of this
pious and affectionate mother, that we may trace the inflexible
integrity, the unwavering love of truth, and the high moral
principle, for which he was through life distinguished. From her
he learned his first lesson as to the value of truth. While a child,
playing behind his mother's chair, he had, unobserved by her,
unrolled a ball of yarn, from which she was knitting, and involved
it in inextricable confusion. When she discovered the mischief,
and addressed him with some severity of manner, he denied having
done it. She at once entered into a very serious conversation
with him, and while she told him that the original matter of
offence was but trifling, she explained to him so fully the
meanness and criminality of falsehood, and urged him with
so much earnestness never again to be guilty of it, that this lesson
of his infancy became indelibly impressed upon his heart. He
was always a favorite child. She was interested in the early
development of his character and talents, and, it is said, was
sometimes obliged even to restrain and check his fondness
for study, as being excessive. She has been heard to say that he
would be " something decided." To use his own expression, " she
idolized him ; " and he never spoke of her but in terms of the
most respectful and affectionate remembrance. Her excellent
influence extended to all her children. They grew up together,
as one observed of them, " a loving household," remarkable
VOL. IV. d
14 MEMOIR.
as a family for their excellent moral character and their strong
mutual attachment.
Dr. Bowditch survived all his brothers and sisters for nearly
thirty years. The eldest, Mary, born March 27, 1766, took upon
herself, after her mother's death, the whole charge of the family,
and almost supplied, towards the younger children, the place of
that excellent parent whom they had lost. She married, April
20, 1791, Mr. David Martin, a ship-master, who died a few years
afterwards. She herself died December 2, 1808, at the age of
forty-two years, leaving to her brother's care an only child, who
afterwards always resided in his family, repaying his kindness
by the attentions of a daughter, and being to us as an elder
sister. As such, she unites with us upon the present occasion.
The second child, Habakkuk, was born May 2, 1768, and
was drowned in Boston about forty years ago. It is not known
that he manifested any peculiar taste for the study of mathematics.
The third child, Elizabeth, was born May 16, 1771, and died
December 9, 1791, in consequence of a fall down a flight of
steps, having lingered in great agony a short time after the
accident. Dr. Bowditch often mentioned, with much emotion, the
circumstances of the death of this sister, who, being but two
years older than himself, had always been the object of his
peculiar regard and affection. In the midst of life and health,
and with a countenance radiant with smiles and joy, she was
about to leave her friends for a few moments, when a single
misstep removed her from them forever.
MEMOIR. 15
The fifth child, William, was born May 5, 1776. He
embraced a sea-faring life, and, while absent upon one of his
voyages, died at Trinidad, in the autumn of 1799, at the age
of twenty-three years. He was quite as remarkable for his
mathematical talents as his elder brother, and, had he lived,
might perhaps have been equally distinguished for the successful
cultivation of this branch of science. In the first edition of the
Practical Navigator he is mentioned as the author of one of
the notes to Table XIV. Dr. Bowditch delighted to speak
of the purity, and almost holiness of character of his brother
William ; and another, who knew well his early virtues, has
said of him, " He was sanctified from his birth."
The sixth child, Samuel, was born September 13, 1778, and
died April 5, 1794, at the age of sixteen years. He also possessed
great quickness at mathematical calculations. But he pursued
his studies with a waywardness and eccentricity which would
probably have prevented his acquirements from being as great
as might have resulted from a more regular and systematic
cultivation of his naturally excellent talents.
The seventh child, Lois, was born March 29, 1781, married
her cousin, Mr. Joseph Bowditch, September 28, 1806, and died
without children, July 29, 1809, aged twenty-eight years. The
eldest and youngest daughters only of this family were married,
and they were also both on their death-beds at one time and place,
prostrated by the same fatal disease, consumption.* Dr. Bowditch,
* One of them, at this time, presented to the eldest child of Dr. Bowditch a little silver
16 MEMOIR.
in his last illness, said that it had always been a source of pleasure
to him to remember that these sisters, when dying, had each told
him that he had been through life a good brother to them;
" but," added he, ** it gives me greater pleasure now than ever
before." He also said that " they died with the calmness of two
Stoics." He once mentioned that, in settling the estate of his
deceased sister, the Judge of Probate thought that he had
discovered a mistake in the fact that the estate had not been
represented insolvent, although more money had been paid away
than had been received. The matter was explained, however,
by the statement of the administrator, that he should have felt
himself disgraced by leaving undischarged the few small debts
incurred by his sister, chiefly during her last illness, while he
possessed any means of his own with which to pay them.
Such were the parents and such the brothers and sisters
of Dr. Bowditch ; and amid the domestic influences which have
been described were the years of his infancy and childhood passed.
Many amusing and interesting incidents of this period of his life
might be mentioned. It was one to which he himself always
recurred with pleasure, as having been very happy, notwithstanding
its many privations. If he was obliged, from motives of economy,
to wear the thin garments of summer when the near approach of
winter made them less comfortable, he would reply to the laugh
of his schoolmates or playfellows by charging them with effeminacy
for preferring warmer clothing. If, as was often the case, he sat
medal, bearing upon it their names, and the inscription, "Virtue and Religion lead to
Happiness." Such had been the result of her own experience.
maioiR. 17
down to a dinner consisting chiefly of potatoes, he felt that a
mealy potato was as good fare as he desired. He humorously
described one occasion, upon which, when sent to buy a warm
loaf of bread for breakfast, he found himself unable, on his way
home, to resist the temptation of gradually eating out the soft part,
so that, upon his arrival, the upper and under crusts had come in
contact. Possessing health and activity of body, he engaged at
one moment with earnestness and ardor in all the amusements
of boyhood, and in the next returned with increased pleasure to
his studies. Yielding sometimes to the impulse of the moment,
as in the instance last cited, he committed trifling indiscretions,
but nothing mean or vicious was ever developed in his character.
Blessed with a joyous and happy temper, he contentedly
accommodated himself to the necessities of his situation. The
son of a poor mechanic, with no expectations from family or
friends, he had within him an energy of purpose by which he
was finally to surmount all obstacles.
While he was yet in his infancy, his father removed with
the family from Salem to the adjoining village of Danvers, and
resided there several years. The house which he occupied is still
standing. It is a humble cottage. The main building, as seen in
front, exhibits but one door and one window. It was here that
his mother first showed him the slight crescent of the new moon,
and the fuller orb of the harvest moon, and perhaps first awakened
in his mind a curiosity to know more of the nature and laws
of the planetary system. He here received instruction from a
school-mistress, whose aged relatives still live in the immediate
vicinity, and by whom it is distinctly remembered that he was
VOL. IV. e
18 MEMOIR.
" a likely, clever, thoughtful boy ; " that " his instructress took
mightily to him ; " that ** he was the best scholar she ever
had ; " that " he learnt amazing fast, for his mind was fully given
to it ; " and that *' he did not seem like other children ; he
seemed better,"
Upon the return of the family to Salem, he was sent, at the
age of seven years and three months, to the best school in the
town, kept by a Mr. Watson. The character of this " seminary
of learning " may perhaps be better realized from the following
circumstances than from any more general or elaborate description
of it. There was but one dictionary in the school ; and a
gentleman, who was a fellow-pupil with Dr. Bowditch, never
saw one while he was there. Each day, the scholars were
called upon to spell aloud, all together, in chorus, the word
honorific abilitudinity ;* spelling and pronouncing the first syllable,
then the two first, three first, &c., which process, applied to the
whole word, of course occupied several minutes. He early
showed a great fondness for mathematics ; but, on account of
his extreme youth, his master, it is said, refused to permit him
to enter on that branch of study until he had obtained and
produced from his father a special request to that effect. He,
upon one occasion, solved a problem in arithmetic, which the
instructer thought must be above his comprehension, and therefore
charged him with having procured the assistance of some older
scholar, giving him a severe reprimand for the attempt to deceive
him. The timely interference and explanations, indeed, of his
* This word, meaDing honor, may be found in Bailey's English Dictionary; and
Shakspeare uses honorificabilitudinitatibus, in Love's Labor Lost, Act 5, Scene I.
MEMOIR. 19
eldest brother, saved him from any actual chastisement ; but this
indignity and act of injustice Dr. Bowditch never forgot.
But even the slight elementary instruction which he might
have obtained at this school, he was obliged to forego altogether
at the age of ten years and two months, when he was taken by
his father into his cooper's shop, that he might by his labor assist
in the support of the family. After remaining here a short time,
he entered as a clerk or apprentice into the ship-chandlery shop of
Messrs. Ropes and Hodges, when he was about twelve years of
age. In this shop he remained till his employers retired from
business, at which time, as early as 1790, he entered the similar
shop of Mr. Samuel Curwin Ward, where he remained until he
sailed on his first voyage, in 1795. Here, when not engaged in
serving customers, he spent his time in reading, and particularly in
the study of mathematics, for which he then felt a confirmed and
decided taste. Upon one occasion, a visiter entered the shop, and,
looking at the two clerks, one of whom was asleep behind the
counter, and the other diligently occupied with his slate and
pencil, smiled and said, " Hogarth's apprentices ! " Another visiter
observed that, if he kept on ciphering so, he had not any doubt
that, in time, he would become an almanac maker! And in fact,
in the year 1788, he computed an almanac for the year 1790,
which is still preserved in his library, being one of the most
curious, if not most valuable, manuscripts in that collection. It
is also stated that he occasionally tried his dexterity at
philosophical experiments ; one instance mentioned being that,
while in the shop of Ropes and Hodges, he constructed quite a
curious barometer. There is now in his library, also, among
20 MEMOIR.
other similar articles, a very neat wooden sun-dial, which he
made in the year 1792.
These pursuits were, however, only the amusement of his
leisure hours. He never allowed them to interfere with the
discharge of his duty towards his employers. Upon one occasion,
indeed, a customer called and purchased a pair of hinges, at a
time when the young clerk was deeply engaged in solving some
problem in mathematics, which he thought he would finish
before charging the delivery of them upon the books, and when
the problem was solved, he forgot the matter altogether. In a
few days, the customer called again to pay for them, when Mr.
Hodges himself was in the shop. The books were examined,
and gave no account of this purchase. The clerk, upon being
applied to, at once recollected the circumstance, and the reason of
his own forgetfulness. From that day, he made it an invariable
rule to finish every matter of business which he began, before
undertaking any thing else. Upon his recommendation, given
quite late in life, one of his sons adopted as a motto for a seal,
" End what you begin." He has himself more than once said,
that he never forgot the hinges.
Having once heard, in 1787, from his brother William, a vague
account of a method of working out problems by letters, instead
of figures, he succeeded in borrowing the book which contained
it, and was so much interested and excited by his first glance
at algebra, that he could not get the least sleep during the whole
of the next night. An old British sailor, residing at Salem on half
pay, and who ended his days as an inmate of the Greenwich
21
Hospital, taught him the elements of navigation ; and when they
last met, as he was about to embark for Europe, he patted him on
the head, saying, " My boy, you have a taste for these things :
keep on studying, and you will be a great man yet:" — an approval
which greatly stimulated and encouraged him. He rose each
day at the earliest dawn, and devoted his morning hours to study.
He has often been heard to say, that the time which he thus
gained from sleep, gave him, substantially, all his mathematics. He
passed the long winter evenings, too, by the kitchen fireside of his
employer, Mr. Hodges, — which his diffidence, as well as the security
it offered him from interruption, led him to prefer to the parlor, —
quietly engaged in his favorite pursuit ; and occasionally, it is said,
also rocking, at the same time, the infant's cradle, at the request
of the attendant, who wished to be doing something else.
It happened that Mr. Hodges and another gentleman owned
together in moieties a very irregularly-shaped field in Salem,
and wished to divide it. Accordingly, the young apprentice
undertook to make the proposed survey and division, and
completed the task with the most minute accuracy. The
co-tenant, however, refused to abide by this survey, since he
thought that, as it was made by one who was in the employment
of Mr. Hodges, it was probable that there had been an unfair bias
in his favor. A regular surveyor was then employed, and Dr.
Bowditch, who was very indignant at the suspicions entertained
in regard to his own result, said that he could not help feeling a
malicious pleasure when he found that the gentleman alluded to
received for his half part several square feet less than he was
entitled to. In 1794, he was employed by the town to assist
VOL. IV. /
22 MEMOIR.
Captain John Gibaui in making a survey of Salem, which labor
he accordingly performed ; and the exact area of the town, as
ascertained by this survey, was computed by him.
Being very fond of books, and having no guide in the selection
of them, his reading in early life was of the most miscellaneous
character. Thus he read through the whole of Chambers's
Encyclopaedia, in four folio volumes, without omitting an article ;
and, as his memory, except as to persons and names, was
wonderfully retentive, he in this manner acquired a fund of the