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Chapter 5


Histories make men wise; poets witty; the mathematics subtile; natural philosophy deep; moral grave; logic and rhetoric able to contend.
--Francis Bacon

When he was a young man Francis Bacon said, "I take all knowledge to be my province." That statement has been
criticized as being boastful.

But in his day there existed very little knowledge of a precise character. He was speaking, I think, with a reverence
for the mysteries of Nature and expressing a wish to discover her ways.

In the Sixteenth Century, anyone looking for a reliable answer to a scientific question would be told to consult
Aristotle who had a theory to explain every physical event. This Greek was often wrong but he was right, often
enough, to be venerated in such glory that hardly anyone dared question his two thousand-year old opinions. Not
even the English Reformation had dented his prestige in the universities. In Spain the Inquisitors were still
searching for those so foolish as to flout Aristotle's authority.

Bacon, after two and a half years at Trinity College, Cambridge, quit in disgust. Forever after he protested against
the "schoolmen" who taught almost nothing but ancient languages, literature, Euclid, Aristotle and religion. They
had a contempt for science, as we understand the term. They believed only in deductive reasoning, thus attempting
to leap over the physical evidence or tunnel under it or find a handy detour around it.

The deducer travels from some premise, which he may have made up all by himself, to a conclusion. In the
Sixteenth Century many religious dogmas embraced such premises; they were regarded as heresies by the Church
of Rome. Before Bacon science itself was influenced by superstition. Alchemy and astrology were then considered
to be themselves sciences; nevertheless, such studies were actually the foundations for modern chemistry and

Before Bacon there was no organized way of thought, so as to permit the accidental discovery of some unknown
phenomenon to be studied, communicated, replicated and refined. The scientific method was practiced, if at all, by
only a few quiet and rare individualists working in private investigation.

Bacon's invaluable contribution to wisdom was his advocacy of inductive reasoning, the common-sense way of
finding a general rule from a number of experiences. This may not seem like much to us now but in his day it was a
revolution, a radical extravagance in thinking. Hardly anyone had then considered experimentation, or observation,
or making improved instruments and recording readings, or simplifying arithmetical computation, or repeating
someone else's experimental process precisely, or writing papers for journals, or communicating with others in the
same field. That was hard work and required inspiration and time and professional devotion. It also took money
and that was scarce. The need was not perceived by government or industry.

Bacon believed that such labors would improve the condition of mankind. He wrote persuasively and he lectured
and he published his views again and again. He wrote in a language then called "vulgar" (English) but he translated
many of his works into Latin, the universal language of that age; they were read in that and many other languages
to which they were retranslated.

In Francis Bacon's Sylva Sylvarum , or A Naturall History , 1627, he reported a thousand "experiments." This was
his attempt to collect, not necessarily from his own experience but from the reports of others, instances of the
behavior of Nature. In reading it, one must remember that no one in England had previously made such a
collection and that there were no proven standards by which to judge the merit of any observation or experiment.
A prize possession of mine is a 1628 edition of Bacon's Sylva Sylvarum . It is a very sturdy book or it would not
have lasted this long. Made of the finest, hand-made rag paper and bound in its original calf, it has seen a lot of
history: Cromwell's rebellion, the Restoration, the Plague, the Black Death, the Great Fire of London, constant war
either with France or Spain, the American Revolution, Napoleon--through it all the succeeding owners somehow
preserved this book.

There are a few waterspots and missing endpapers; it is not the first edition but the second. The frontispiece went
into the press under the original engraving, except that someone has made a 9 out of the 7 in the date. It has had at
least one proud owner with a well-practiced hand; he has added to his script many flourishes:

                                                                              Mr. John Rainor
                                                                                His Naturall
                                                                To Mr. Raynor (sic) I doe belong
                                                                  In keeping mee you doe Rong.
                                                                   Return me to h hand Again
                                                              And hel Requite you for your Paine.

Next is the engraved portrait of "The right Hon\ble\ Francis Lo. Vervlam, viscount S\ct\ Alban. mortuus 9 Aprilis,
Anno Dni. 1626. Annog Aetat 66 ." His family motto, MONITI MELIORA is shown, along with his crest and MEDIO
CRIA FIRMA . Facing that is the frontispiece, an engraving of the globe, marked Mundus Intellectualis , hovering
between two pillars which perhaps represent the Pillars of Hercules. One wonders, how many modern books will
be worth keeping in such readable condition and still exist in A.D. 2348?

Bacon divided his book into ten "Centuries," each division containing 100 reports. In Century I, experiment 33, Sir
Francis Bacon says:

"It is affirmed constantly by many, as an vsuall Experiment; That a Lump of Vre , in the Bottom of a Mine, will be
tumbled, and stirred, by two Mens strength; which if you bring it to the Top of the Earth , will aske Six Mens
strength at the least to stirre it.It is a Noble Instance , and is fit to be tried to the full: For it is very probable, that the
Motion of Gravity worketh weakly, both farre from the Earth, and also within the Earth: The former, because the
Appetite of Vnion of Dense Bodies with the Earth, in respect of the distance, is more dull; The latter because the
Body hath in part attained his Nature, when it is some Depth in the Earth. For as for the Morning to a Point or place
(which was the opinion of the Ancients ) it is a meere Vanity."

Did Sir Isaac Newton (1642-1727) ever read that? He did, of course, first discover the mathematical laws governing
gravity. His book, Philosophiae Naturalis Principia Mathematica of 1687 is considered one of the greatest
contributions to the science of motion and the study of gravitation. Bacon's story about the weight of a body in a
mine may be apocryphal, but his conclusions are weirdly and accurately perceptive. Newton would agree that the
weight of a rock, measured by a spring balance, would decrease as its distance from the earth increased and that its
weight, if within the earth, would decrease as it moved nearer to the center of the earth's gravity. Had there been
private discussions of these principles in Bacon's time? Had these primal notions been floated about for almost a
hundred years afterward, until they struck and inspired Newton's mind?
In Century I, experiment 100, Bacon says:

"There is nothing more Certain in Nature, than that it is impossible for any Body , to be vtterly Annihilated ; But
that, as it was the work of the Omnipotency of God , to make Somewhat of Nothing ; So it requireth the like
Omnipotency, to turn Somewhat into Nothing . And therefore it is well said, by an Obscure Writer of the Sect of the
Chymists ; That there is no such way to effect the Strange Transmutations of Bodies , as to endeuour and vrge by
all means, the Reducing of them to Nothing ..."

What Bacon is saying here is a preview of the first law of atomic physics, the law of conservation. This is that
neither matter nor energy may be created or destroyed. One may be changed to another, it has recently been
discovered, but neither of these elemental things may be utterly annihilated. In this passage he also disparages the
alchemists and their attempts to change lead into gold.

Bacon has been criticized because, it is said, he was not a scientist. Of course, science has now become a way of
working. If his inductive demands had not, in his time, yet been adopted, then there were no contemporary
scientists. It has been said that he was not a mathematician and that he contested the proofs of Copernicus that the
earth circuited the sun. European mathematics was then itself in its infancy, while ordinary calculations presented
great difficulties and the chances of error were multiple. Copernicus' theory was filled with faults and was greatly
improved by Johannes Kepler by the later publication, in Central Europe, of his three laws of planetary motion. But
that was not soon enough for English scientific opinions to alter.

At the time long division was done by the "scratch" method invented by Fibonacci about 1202. Here is an example
of his way of dividing 65284 by 594:

Math aficionados may wish to try to determine how the answer, 109, was arrived at. And the answer, of course, is
wrong. It should be 109.90572+, as our little pocket calculators can instantly compute. But there was not, published
in England until late in Bacon's life, even the concept of the decimal system. There were no log tables to immensely
reduce calculating time and error in the science of astronomy. There was really no handy way to check up on
Copernicus and, until there was, maybe Bacon reached the correct scientific conclusion: not adequately proven. Not
until Napier, Stevin and Briggs who made logarithmic tables and instructed the scientific community in the decimal

J. G. Crowther, author of 35 books on the history of science, has this to say about Francis Bacon's understanding of

"Bacon regarded mathematics as belonging to what he called metaphysics or the principles of nature. It was the
auxiliary or handmaid to every branch of science. But it had come to pass, he knew not how, that mathematics and
logic had presumed to domineer over science on the strength of their certainty. Profound though it is, mathematics
should be kept in its place. He suggested that there was room for improvement in 'the abridgment of compilation,'
and in the use of infinite series in physics. Napier's invention of logarithms, the invention of the calculus, and of
modern calculating machines bear him out.

"He predicted that as physics 'advances farther and farther every day,' it will 'require fresh assistance from
mathematics in many things. ... If men be not idle, many new branches of applied mathematics will come into
existence... The inquiries will have the best result when they begin with physics and end with mathematics' " [@].

Patent laws did not exist, and Bacon counseled:

"If any man out of his own wit, industry or endeavour, find out anything beneficial to the Commonwealth, or bring
any new invention which every subject of this kingdom may use; yet in regard of his pains and travel therein, her
Majesty perhaps is pleased to grant him a privilege to use the same only by himself or his deputies for certain time.
. .If a man could succeed not in striking out some particular invention, however useful, but in kindling a light in
nature. . .and bring into sight all that is most hidden and secret in the world,--that man (I thought) would be the
benefactor indeed of the human race,--the propagator of man's empire over the universe, the champion of liberty,
the conqueror and subduer of necessities."

In these thoughts Bacon went beyond the principle of the government-bestowed patent; he was suggesting basic
research as the duty of the new scientists. Later he proposed that basic research, being both expensive and
neglected, should be supported by the government. Yet then and now many have regarded him as a promoter of
applied science, rather than of the "pure science" which his detractors somewhat piously profess.

Loren Eiseley was a famous anthropologist who found new and much earlier dates for the fossils of the Pleistocene
Epoch. Among such remains were specimens of the bones of humans, or humanoids, more than a million years old.
He was fascinated by Bacon and wrote prose poems in his praise. In a chapter entitled "The Man Who Saw Through
Time [@]," he speaks of his accomplishments:

"It is not possible to realize the full magnitude of Bacon's achievement without some knowledge of this age of the
scientific twilight--an age when men first fumbled with the instruments of science yet, in the next breath, might
consider the influence of stars upon their destinies or hearken to the spells of witchcraft. . .Not all men, like Sir
Francis Bacon, are fated to discover an unknown continent, and to find it not in the oceans of this world but in the
vaster seas of time. Few men would seek through thirty years of rebuff and cold indifference a compass to lead
men toward a green isle invisible to all other eyes. . .Appropriately there lingers about this solitary time-voyager a
shimmering mirage of fable, an atmosphere of mystery, which frequently closes over and obscures the great
geniuses of lost or poorly documented centuries.

"It is in the use to which he put inductive logic that he strove to break out of the old, unproductive circle of the
Aristotelian schoolmen. In essence his argument is as follows: We must refrain from deducing general laws or
principles for which we have no real evidence in nature. Instead, because of our human tendency to leap to
unwarranted conclusions, we must dismiss much of what we think we know and begin anew patiently to collect
facts from nature, never straying far from reality until it is possible through surety of observation to deduce from
our observations more general laws. . .

"Yet Bacon, for all his emphasis on observation, was ahead of his time and writes, indeed like a modern theoretical
physicist, when he argues that "many parts of nature can neither be invented, that is observed, with sufficient
subtlety, nor demonstrated with sufficient perspicuity without the aid and intervening of the mathematics."

Eiseley continues and quotes Bacon's "draft" will (it was a later one that was offered for probate):

" 'I leave my name,' wrote Francis Bacon, Lord Chancellor of England under James the First, 'to the next ages, and
the charity of foreigners, and to mine own countrymen after some little time be passed.' Men like Bacon are not
easily loved or used: something terrific exists in them, however humbly they speak... 'Even to deliver and explain
what I bring forward,' Bacon once remarked in weariness, 'is no easy matter, for things in themselves new will yet
be apprehended with reference to what is old.' In the passage of long centuries the endless innovations of science
have not quieted that lust for power which still blocks the doorway to the continent of Bacon's dreams...

" 'The unlearned man,' wrote Bacon carefully, 'knows not what it is to descend into himself, or call himself to
account...whereas, with the learned man, it fares otherwise that he doth ever intermix the correction and
amendment of his mind with the use and employement thereof.' ...

"Bacon himself, perhaps out of bitter self-questioning and disappointment, referred to the world he inhabited as
one of shadow rather than of light...

"I have said that his building is huge beyond our imaginations, drafty and unfinished. Like all such monuments of
genius, it is never truly of the past. Lights flicker mistily in its inner darkness, stones are still moved about by
unseen hands. Somewhere within, there is a ghostly sound of hammering, of a work being done. The work is ours,
the building is as we are shaping it, nor would Bacon have it otherwise. Since Bacon was a statesman and a
pathfinder, no man quite escapes his presence in the haunted building of science, nor the whispers of his
approbation or unease.

"Occasionally the voice grows louder, as now [1962], in our overtoppling part of the structure. To those who listen,
the harsh Elizabethan line strikes once more like surf around the shores of his far-off New Atlantis, warning us of
man's double nature and perhaps his fate, for Bacon did not hesitate to write: 'Force makethe Nature more violent
in the Returne.' "

Eiseley might have agreed with Crowther who said, "Bacon was an autodidact who thought everything out for
himself. When this kind of man possesses great mental power his work has a perpetually stimulating originality,
because it owes exceptionally little to conventional ideas..."

These were the supernal thoughts growing in Bacon's mind almost four centuries ago; these, while raw sewage still
flowed through the open gutters of the London streets.

John Napier was born in Edinburgh, Scotland in 1550.

The spelling of his name is of particular consequence, as will be shown in a later chapter. Webster's Biographical
Dictionary gives it as "Napier" or "Neper," as do other books of reference. Another spelling is shown on the
title-page of Mirifici Logarithmorum , printed in Edinburgh in 1614. This is "Nepero."

On the title-page of Arithmetica Logarithmica , published by Gvlielmvs Iones in London in 1624, the name is
spelled "Nepervs." In John Speidell's New Logarithmes , London 1624, the name is "Nepair." The orthography of
Napier's name was then quite variable which may offend the modern eye, but it is clear that it was spelled
auricularly, much as it sounded.

A Scottish nobleman, he was the eighth Laird of Merchiston. He had studied at St. Andrews college and on the
continent. He had a religious bent and wrote a defense of Protestantism in 1593 [@]; this became very popular and
was translated into several languages. He lived in a castle at Gartness and fathered twelve children. Somewhat of a
polymath, he studied agriculture and promoted the use of manure and ordinary salt as fertilizer. He invented a
hydraulic screw machine to pump water out of the coal pits. He also suggested designs for armaments, such as
burning mirrors for setting fire to ships, artillery, and an armored chariot built so that its occupants could fire in
any direction. He showed an early interest in mathematics by writing, in 1572, a treatise on arithmetic and algebra.
This work showed that he knew something about the imaginary roots of equations, which was rare in those times.
He also invented a device known as "Napier's bones"; this was an early calculator, a sort of mechanical
multiplication table made of square sticks and operated by manipulation and inspection. Because of his many
talents he was once accused of being a Wizard.

In the Dictionary of Scientific Biography [@], it is said that in the last section of Napier's 1617 Rabdologiae he
described another device, "a mechanical method of multiplication that was based on an `areal abacus' consisting of
a checkerboard with counters in which numbers were expressed in the binary scale . . .The calculation of the canon
was a tremendous task and occupied Napier personally for over twenty years." It is known that he began work on
logarithms and the tables about 1590.

However painful it may be to those far removed from high-school math, some explanation of logarithms should be
offered from a text [@].

The logarithm is defined as the exponent of a base number raised to a power. Thus, if

                                                      then b is the logarithm of c to the base a, or

                                      Powers of the same base are multiplied by adding the exponents:

                 Therefore, the logarithm of a product is the sum of the logarithms of the factors. Thus

There are two standard systems of logarithms: the [early] Napierian, or natural, system, with the base e = 2.71828+
and the Briggs, or common, system with the base 10. The former is generally used in algebraic, the latter in
numerical calculations. ... The tables are useful for performing repeated multiplication or divisions, when greater
than slide rule accuracy is desired.

By using logarithms we substitute the simple process of addition for the more involved process of multiplication.
Instead of multiplying two numbers together, as

                                                                           123 X 456 = 56,088

we add their logarithms, and look up the corresponding number in a log table:

                                              log 123 + log 456 = 2.08991 + 2.65896 = 4.74887 = log 56,088

And, of course, Napier's invention was also used for division, to determine square and other, higher order roots,
and especially in trigonometry and geometry. He made the tools for astronomers to study the planets in their
travels and to determine why they sometimes appeared to follow retrograde epicycles, to temporarily reverse their
courses while orbiting the sun.

Napier must have communicated with Henry Briggs (1556-1631) before the publication of Mirifici Logarithmorum
Canonis Descriptio in 1614, though they did not meet until a year later. Briggs was a professor of geometry at
Gresham college in London; he had graduated from Cambridge in 1581.

Briggs wrote to Archbishop Usher in March of 1615, "Napper, lord of Markinston, hath set my head and hands at
work with his new and admirable logarithms. I hope to see him this summer, if it please God, for I never saw a
book which pleased me better or made me more wonder." In a life of Briggs, Dr. Thomas Smith says of his regard
for Napier's Canon Mirificus , "He cherished it as the apple of his eye; it was ever in his bosom or in his hand, or
pressed to his heart, and, with greedy eyes and mind absorbed, he read it again and again. ... It was the theme of
his praise in familiar conversation with his friends, and he expounded it to his students in the lecture room."
In the summer of 1615 Briggs traveled from London to Edinburgh. William Lilly, an astrologer, wrote of his
meeting with Napier at Merchiston Castle:

"I will acquaint you with one memorable story related unto me by John Marr, an excellent mathematician and
geometrician whom I conceive you remember. He was servant to King James I and Charles I. When Merchiston
first published his Logarithms Mr. Briggs, then reader of the astronomy lectures at Gresham College in London,
was so surprised with admiration of them that he could have no quietness in himself until he had seen that noble
person whose only invention they were. He acquaint John Marr therewith who went into Scotland before Mr. Briggs
purposely to be there when these two so learned persons should meet. Mr. Briggs appoints a certain day when to
meet at Edinburgh; but, failing thereof, Merchiston was fearful he would not come. It happened one day as John
Marr and the Lord Napier were speaking of Mr. Briggs. 'Oh! John,' saith Merchiston, 'Mr. Briggs will not come
now'; at the very instant one knocks at the gate. John Marr hasted down and it proved to be Mr. Briggs to his great
contentment. He brings Mr. Briggs into my Lord's chamber, where almost one quarter of an hour was spent, each
beholding the other with admiration, before one word was spoken. At last Mr. Briggs began, 'My Lord, I have
undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you
came first to think of this most excellent help unto astronomy, viz. the Logarithms; but, my Lord, being by you
found out, I wonder nobody else found it out before, when, now being known, it appears so easy.' "

They spent a month together and Briggs came for another visit in the following year. He planned to return again
until he received news of Napier's death.

Alfred Hooper, in his Makers of Mathematics [@] has this to say about the invention of the decimal system:

"Decimal fractions are the most important development of arithmetic since the introduction of Hindu-Arabic
number-symbols. They enable parts of a whole to be added, subtracted, multiplied and divided, etc., as whole
numbers, and thus avoid the clumsy and complicated methods involved in handling other fractions. Their use was
first clearly advocated by a mathematician named Simon Stevin of Bruges, better known as Stevinus, in a work
published in 1582 and called La Practique d' Arithmetique . Many previous mathematicians had almost, but not
quite, hit on the idea of decimal fractions. For instance, tables of square roots had been drawn up for numbers
which had first been multiplied by 1,000,000. The roots as given in the table were, of course, 1,000 times too great,
but by this method it was possible to avoid the use of fractions, at least for approximate values of the roots. We
have seen the somewhat similar method adopted by compilers of the values of trigonometric functions. The idea
that lies behind our present 'decimals' was only gradually reached, the process of thought involved in its
development covering hundreds of years...

"Napier seems to have been the first writer to use a period to mark the end of the whole numbers, and to realize
that the decimal fractions occupied places which could be regarded as lying on an extended abacus, to the right of
the units' wire. In the Constructio , Napier said, 'In numbers distinguished by a period in their midst, whatever is
written after the period is a fraction, the denominator of which is unity with as many ciphers [zeros] after it as there
are figures after the period...In computing tables, these large numbers may again be made still larger by placing a
period after the number and adding cyphers...' Here is yet another example of a very simple idea that was to have
tremendous consequences. In this connection, it must be remembered that the Constructio was not published until
two years after Napier's death. So it is impossible to say with certainty whether these sentences were inserted by
Napier or by Briggs, who revised the work before publication. The fact that Napier does not use a decimal point or
its equivalent in his Descriptio seems to indicate that this simple yet fruitful invention is due to Briggs. On the other
hand, it is difficult to imagine how Napier's calculations...could have been carried out without the use of the
decimal point.

Napier invented natural logarithms and at some time after 1590 he suggested to Henry Briggs changing the system
so that the logarithm of unity would be zero. Briggs worked out the tables for the "common" system, but credits
that idea to Napier in the preface to his Arithmetica Logarithmica of 1624.
In his Descriptio of 1614 Napier says, "It was indeed left at libertie in the beginning, to attribute nothing, or 0, to
any sine or quantitie [for its logarithm]." And Briggs says, in the Arithmetica Logarithmica of 1624, "I my
auditors in Gresham College, remarked that it would be much more convenient that 0 should be kept for the
logarithm of the whole sine. ... And concerning that matter I wrote immediately to the author himself [Napier]; and
as soon as the season of the year...permitted I journeyed to Edinburgh, where....he said that he had for some time
been of the same opinion...that the change should be...that 0 be the logarithm of unity..."

Professor George A. Gibson, M.A., L.L.D. sums up in a "Handbook" [@] for the Napier Tercentenary Celebration:

"At the first visit Napier and Briggs discussed certain changes in the system of logarithms. In a letter to Napier
before the first visit, Briggs had suggested that it would be more convenient, while the logarithm of the whole sine
was still taken as zero, to take the logarithm of the tenth part of the sine as a power of 10, and he had actually
begun the calculation of tables of his proposed system. Napier agreed that a change was desirable, and stated that
he had formerly wished to make a change; but that he had preferred to publish the tables already prepared as he
could not, on account of ill-health and for other weighty reasons, undertake the construction of new tables. He
proposed, however, a somewhat different system from that suggested by Briggs, namely, that zero should be the
logarithm, not of the whole sine but of unity, while, as Briggs suggested, the logarithm of the tenth part of the sine
should be a power of 10. Briggs at once admitted that Napier's method was decidedly the better, and he set about
the calculation of tables on the new system, which is essentially the system of logarithms now in use. "

So, the general idea had occurred to them both; Napier had first recommended that logarithms be calculated to the
base 10 in which the log of 1=0, and the use of some form of decimal fractions was required for its application.
In 1619 there was imprinted at London a description of the construction of Napier's tables with the title Mirifici
Ipsius Canonis Constructio . In the appendix appeared this statement:

"On the construction of another and better kind of Logarithms, namely one in which the Logarithm of unity is 0. . .
Among the various improvements of logarithms, the more important is that which adopts a cypher as the
Logarithm of unity and 10 000 000 as the Logarithm of either one tenth of unity or ten times unity. Then these being
once fixed, the Logarithms of all other numbers necessarily follow."

Simon Stevin's French treatise, De Thiende of 1585, was translated by Robert Norton and published in London in
1608 under the title of Disme: the Art of Tenths, or Decimall Arithmetike . The name for our U. S. coin, the dime,
has its derivation in the word "Disme."

There is no reason to believe that all of this mathematical activity was a deep secret within the London
philosophical (scientific) community, or to Francis Bacon who lived among them and showed great respect for
mathematics in his published works. It should not be surprising that Bacon in 1609--the year following the
publication of Disme --made, as we shall see, a hidden notation that the expression 00 represented Napier's ciphers.
That was the foundation of the logarithm to base 10 and equaled unity (the number 1). The fact that he did so in a
concealed cipher , and called attention to it with thirty anomalous decimal points, demonstrates the peculiarity and
strangeness of his imagination.



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