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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas

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Isaac Newton's disciples in the eighteenth century
were impressed not only by his discoveries in optics
and celestial mechanics and by his admirably ordered
System of the World, but by the method he employed.
A variety of writers mention him as the inventor of
the only proper way of investigating nature: of a
method that d'Alembert called “exact, profound, lu-
minous and new.” Laplace found his method “happily
applied” in the Principia and the Opticks, works valua-
ble not just for the discoveries contained in them, but
as the best models to be emulated, as the embodiments
of this method (Laplace, pp. 430-31). Moreover,
Newton's method—as understood by men of letters like
Voltaire and Condillac—was thought to be, besides a
technique for investigating physical nature, “a new
method of philosophizing” applicable to all areas of
human knowledge. What Newton called his “Experi-
mental Philosophy” had wide application. It set the
bounds to human presumption; it was systematic, yet
as Condillac pointed out, was opposed to the Esprit
de système;
it rejected unsupported or gratuitous

Analysis, the dissection of nature, men in the eight-
eenth century took to be the key to knowledge, the
great and novel intellectual tool, indeed the essence
of Newton's method. Yet if we read carefully the im-
portant Newtonian passages, or the best of Newton's
expositors, we discover that Newton's methodological
prescriptions were by no means confined to this “dis-
section of nature,” although he lays great stress upon
it. Men who possess Newton's experimental philosophy,
wrote Roger Cotes,

... proceed in a twofold method, synthetical and analytical.
From some select phenomena they deduce by analysis the
forces of Nature and the more simple laws of forces; and
from thence by synthesis show the constitution of the rest.
This is that incomparably best way of philosophizing, which
our renowned author most justly embraced in preference
to the rest

(Newton, Principia, ed. F. Cajori, p. 547).

Colin Maclaurin wrote, “In order to proceed with
perfect security, and to put an end for ever to disputes,
[Newton] proposed that, in our inquiries into nature,
the methods of analysis and synthesis should be both
employed in a proper order” (1775, p. 9).

Newton first referred in print to his methodological
principles, at least in a major work, in the new Queries
added to the Latin version of his Opticks which
appeared in 1706. In one of these Queries (Q. 20/28)
he reproves those “later philosophers” (physici recen-
) who invoke mechanical hypotheses to explain
all things,


[w]hereas the main Business of natural Philosophy is to
argue from Phaenomena without feigning Hypotheses, and
to deduce Causes from Effects

(Opticks, p. 369).

He is even more explicit in a passage towards the close
of the last new Query (Q. 23/31), a passage that was
greatly expanded when Newton in 1717-18 brought
out a second English edition of his Opticks. Since this
English text is the most familiar, to say nothing of being
more complete, and indeed is the locus classicus for
any study of Newton's method, it deserves to be given
here in full, with the passages that had earlier appeared
in the much shorter Latin statement of 1706 given in

As in Mathematicks, so in Natural Philosophy, the Investi-
gation of difficult Things by the Method of Analysis, ought
ever to precede the Method of Composition. This Analysis
consists in making Experiments and Observations,
and in
drawing general Conclusions from them by Induction, and
admitting of no Objections against the Conclusions, but such
as are taken from Experiments, or other certain Truths. For
Hypotheses are not to be regarded in experimental Philoso-
phy. And although the arguing from Experiments and Ob-
servations by Induction be no Demonstration of general
Conclusions; yet it is the best way of arguing which the
Nature of Things admits of, and may be looked upon as
so much the stronger, by how much the Induction is more
general. And if no Exception occur from Phaenomena, the
Conclusion may be pronounced generally. But if at any time
afterwards any Exception shall occur from Experiments, it
may then begin to be pronounced with such Exceptions
as occur. By this way of Analysis we may proceed from
Compounds to Ingredients, and from Motions to the Forces
producing them; and in general, from Effects to their Causes,
and from particular Causes to more general ones, till the
Argument end in the most general.
This is the Method of
Analysis: And the Synthesis consists in assuming the Causes
discover'd, and establish'd as Principles, and by them ex-
plaining the Phaenomena proceeding from them, and proving
the Explanations

(Opticks, pp. 404-05).

Several points in this text deserve to be noted. First,
Newton advocates a single procedure, made up of two
“methods” both of which must be employed, but one
of which (the analytic) must be carried out before the
other (the synthetic or compositional). Second, that
although he describes these two methods by terms used
for analogous methods in mathematics, by analysis he
means the making of observations or the performing
of experiments, and deriving conclusions from them
by induction. Experiments are among the certain
truths, yet inductions from them do not “demonstrate”
the conclusions drawn; still this is “the best way of
arguing which the nature of things admits of.”

If this is Newton's most important statement of his
scientific method, what of the famous Rules of Reason-
ing in Philosophy which Newton placed at the begin
ning of Book III of the Principia? With the single
exception of Rule IV, they do not deal with method,
at least as the term was commonly used in Newton's
day, and will be used in this article. The first two
rules—statements of the law of parsimony and of the
principle of the analogy and uniformity of nature—can
perhaps be described as basic articles of scientific faith,
as meta-scientific principles. Rule III, which has been
much discussed and variously interpreted, can be char-
acterized, at least in a loose way, as an analogical rule.
Its manifest purpose is to justify extending observations
and measurements concerning gravity on the earth and
in the solar system so as to “allow that all bodies
whatsoever are endowed with a principle of mutual
gravitation.” Rule IV was added to the third edition
of the Principia (1726) and clearly echoes what he had
written some eight years before in the expanded Query
23/31 of the Opticks. This Rule reads as follows:

In experimental philosophy we are to look upon propositions
inferred by general induction from phenomena as accurately
or very nearly true, notwithstanding any contrary hypothe-
ses that may be imagined, till such time as other phenomena
occur, by which they may either be made more accurate,
or liable to exceptions.

This is clearly a methodological rule which, Newton
adds, “we must follow, that the argument of induction
may not be evaded by hypotheses” (Principia, p. 400).
Newton in his Universal Arithmetick, a work signifi-
cantly subtitled “A Treatise of Arithmetical Composi-
tion and Resolution,” describes arithmetic as synthetic,
since we “proceed from given Quantities to the
Quantities sought” whereas algebra is analytic because
it “proceeds in a retrograde order” assuming the
quantities sought “as if they were given.” And he
comments that “after this Way the most difficult Prob-
lems are resolv'd, the Resolutions whereof would be
sought in vain from only common Arithmetick” (Uni-
versal Arithmetick,
London [1728], p. 1). These remarks
cast light on what Newton had in mind when, in Query
31, he compares the method of investigation in natural
philosophy with those in mathematics.

Method vs. Logic. At this point, let us agree to an
important distinction: that between logic and method.
Both, of course, are concerned with how our mind
should operate in order to arrive at reliable knowledge,
but there are differences. The word method seems to
have been a coinage of Plato; it first appears in his
Phaedrus, where Socrates is advocating an art or
technic of rhetoric as opposed to the devices of the
Sophists. The word suggests a “path” or “route,” being
derived from meta and odos, indicating a movement
according to a road.

To think or argue clearly and effectively it is neces-
sary to understand the route along which we conduct


our thoughts. He who has such a route, such a direc-
tion, possesses method. Logic and method are not the
same thing. Logic is, of course, indispensable to
method: it is the inner machinery conducting us along
the path; it provides us with the tactics we employ.
If method indicates the grand strategy, the road (or
roads) we should follow, logic in turn provides the
means of transportation, together with the code de la

Aristotle was not primarily concerned with the
problem of method as the word is defined here. But
there are passages in the Organon which testify that
he believed there were two modes or directions of
conducting our reasoning: the deductive or syllogistic
and the inductive. Both lead to understanding, and
understanding requires knowledge of the reason for the
fact. But the deductive or syllogistic mode of demon-
stration assumes that we know the cause or principle
from which the consequences can be drawn. In the
jargon of the medieval philosophers, it is demonstration
propter quid (demonstration wherefore) by which we
start from what is prior in the order of nature and end
up with what is “prior in the order of our knowing,”
that is, what is directly accessible to us. Some inquiries
properly move in the opposite direction: from what
is more knowable and obvious to us we proceed to
those things that are “more knowable by nature.” In
the Physics Aristotle makes it clear that in this branch
of philosophy “we must follow this [inductive] path
and advance from what is more obscure by nature, but
clearer to us, towards what is more clear and more
knowable by nature (Physics, I, i, 184a-184b 10).

There is, however, a passage in the Nicomachean
(Book III, Ch. 1, 1112a-1113a 14) where
Aristotle contrasts men who deliberate with those who
analyze in geometry, and men who act upon those
deliberations with those who engage in geometrical
synthesis. This is perhaps the earliest explicit echo of
those two directional ways of conducting thought in
mathematics to which Newton referred in our chief
text (Heath, pp. 270-72). In all likelihood the two
mathematical methods were known in the time of Plato
and Eudoxus. Nevertheless the classical account is that
given much later in the Mathematical Collections of
Pappus who attributes the elaboration of two methods
to the work of Euclid, Apollonius of Perga, and
Aristaeus the Elder (M. R. Cohen and I. E. Drabkin,
pp. 38-39). In analysis, Pappus writes, the mathe-
matician assumes what is sought as if it were true, and
by a succession of operations arrives at something
known to be true. Synthesis, on the other hand, reverses
the process: the geometer starts with what is known
(axioms, definitions, theorems previously proved) and
by a series of deductive steps arrives at what he has
set out to prove. This synthetic method is, of course,
characteristic of the most familiar proofs of Euclid's
geometry. The two methods may be thought of as
alternative paths to be chosen according to the de-
mands of a particular inquiry. Yet it appears likely that
the analytic method is the one used for purposes of
investigation and discovery (as Aristotle implies), and
that this is then followed, for formal demonstration,
by the method of composition or synthesis which,
unlike analysis, follows the “normal” direction of logi-
cal consequence (Hintikka, 1966).

Pappus' text, unknown to the Middle Ages, was
rediscovered in the Renaissance, and it gained currency
especially through Commandino's Latin translation of
1589. The two procedures or methods in mathematics
were obviously common knowledge by Newton's day.
Algebra, in which the sixteenth and the seventeenth
centuries made such notable progress, was seen to ex-
plore problems analytically, although quite differently
from the “geometrical analysis of the ancients.” It is
in this sense that the two procedures are discussed by
Newton's friend Edmund Halley in the preface to his
edition of Apollonius. Algebraic analysis Halley de-
scribes as brevissima simul perspicua (“the shortest and
clearest”); synthesis, by contrast, is concinna et minime
(“elegant and easier”).

There is—and it surely deserves mention—a text
with which Newton must have been familiar. In his
Mathematical Lectures, delivered in 1664-65, Newton's
friend and teacher, Isaac Barrow, equates the mathe-
matical and philosophical uses of these two terms.
Barrow explains why, in enumerating the parts of
mathematics he is “wholly silent about that which is
called Algebra or the Analytic Art.”

I answer, this was not done unadvisedly. Because indeed
Analysis... seems to belong no more to Mathematics than
to Physics, Ethics or any other Science. For this is only...
a certain Manner of using Reason in the Solution of Ques-
tions, and the Invention or Probation of Conclusions, which
is often made use of in all other Sciences. Wherefore it is
not a Part or Species of, but rather an Instrument sub-
servient to the Mathematics: No more is Synthesis, which
is the manner of demonstrating Theorems in Contradiction
to Analysis

(Barrow, p. 28).

The relation between mathematical procedures and
general intellectual method had, we saw, been rather
casually invoked by Aristotle. Yet he devoted most of
his attention to elaborating his demonstrative logic and
his theory of the syllogism. Early in our era—by the
second and third centuries A.D.—philosophers and
commentators began to use the mathematical terms
in writing about method. For example Alexander of
Aphrodisias, in his commentaries on Aristotle, mentions


geometrical analysis as one of nine different senses of
the word analysis used by philosophers.

A most important figure is surely Galen, for he brings
the subject out of the realm of pure dialectic into the
practical world of the physician. His concern is with
method (or perhaps with methods) and with the proper
way the doctor should think about and teach his art.
Galen, we know, wrote a major work called Concerning
but it was subsequently lost. Almost
certainly it did not deal only with logic in the narrow
sense, but with the problem of method, a problem that
arose for him because of the different approaches to
medicine of the chief rival schools into which his con-
temporaries and rival physicians were divided: the
Empiric and the Dogmatic. What he sought was a
middle way between those who relied wholly on accu-
mulated experience, and those who based their proce-
dures upon medical theory. In a treatise called On
Medical Experience
Galen wrote “The art of healing
was originally invented and discovered by the logos
[reason] in conjunction with experience. And to-day
also it can only be practised excellently and done well
by one who employs both of these methods” (R.
Walzer, p. 85). But how are these methods to be

References to method are scattered through Galen's
major works; but his small work, the Ars parva (or
Microtegni)—one of the first Greek medical writings
to be made available in Latin—was the chief vehicle
for transmitting Galen's thoughts on method. It was
translated from an Arabic version into Latin by
Constantine the African in the eleventh century, and
later by Gerard of Cremona who rendered it along with
the remarks of its Arab commentator, Haly Rodohan.
In this form it was printed as an early medical incu-
nabulum. Galen's introductory paragraph is very short,
yet it provided the basis for much subsequent discussion
of method in medicine. Galen says there are three ways
of teaching or demonstrating the art of medicine: these
are by analysis, by synthesis and by definition (Galen,
ed. Kühn, I, 305-07). The use of the mathematicians'
terms “analysis” and “synthesis” may have come to
Galen from philosophical writings, yet it calls to mind
his great admiration for the demonstrative procedures
of the Greek geometers.

Perhaps too much has been made of Galen's
methodology, but it should be emphasized that he is
talking about methods of teaching, of leading the
thought of the learner in teaching him medicine. There
is nothing to suggest that the two procedures are to
be used together, or that they are supplementary
aspects of a single method. Rather it is that medicine
can be taught analytically, by rising from the facts of
observation (as, for example, in anatomy or pathology)
to the principles or causes of health and disease. Or
it can be presented in reverse fashion by starting with
the principles—i.e., with medical theory—descending
thence to the observed facts. But Haly in his preface
identifies these two methods of teaching with the two
directions of reasoning Aristotle presents in the Poste-
rior Analytics:
reasoning that moves from causes to
effects, and that which proceeds from effects up to
causes. (For Haly's prologue see A. Crombie, pp.
77-78. The Latin translation gives conversio and solutio
for “analysis” and compositio for “synthesis.”)

The earliest text in the Latin West involving a dis-
cussion of method is not related to medicine, but to
philosophy. It is a passage in the commentary of
Chalcidius, a fourth-century Christian Neo-Platonist,
on Plato's Timaeus. Chalcidius is discussing the number
and nature of the principles or elements of things; there
is, he says, a duplex probatio for dealing with such
matters, a double method of demonstration, the two
parts of which are called resolutio and compositio,
terms which correspond respectively to analysis and
synthesis. Resolutio is a method of inquiry that begins
with things sensible, prior in the order of understanding
(that is, more known to us) from which we infer the
principles of things, principles which are prior “in the
order of nature.” Compositio (or synthesis) is the
method of syllogistic inference from the principles. The
historian of science, Alistair Crombie, asserts that
Chalcidius “defined the combined resolutio-compositio
as the proper method of philosophical research” (ibid.,
p. 59). But an examination of the Chalcidius text shows
that the business is much more complex: the two pro-
cedures, while in some sense supplementary, do not
seem to form a single method, but are alternative
methods. Resolutio is the method used in arriving at
the material principles of things; compositio has a wider
application: by means of it we demonstrate the formal
relationships (genera, qualitates, figuras) from which
we are led to grasp God's harmonious order and his
providential role (Platonis Timaeus interprete Chalcidio
cum eiusdem commentario,
ed. Iohannis Wrobel,
Leipzig [1876], CCCII-CCCV, pp. 330-34).

Chalcidius' commentary was widely read in the early
Middle Ages. For example we find the terms resolutio
and compositio used by the ninth-century thinker, John
Scotus Erigena, in his mystical De divisione naturae
where the aim is metaphysical understanding. Clearly,
then, resolutio need not be an analysis of natural
phenomena, but an analysis of thought.

The philosophers of the twelfth century had little
interest in problems of method, but rather in logic as
they first found it in the old logic (logica vetus) of
Aristotle. The rationalism, moveover, of an Anselm,
a Gilbert de la Porrée, a Richard of St. Victor, even


an Abelard, led them to deprecate the evidence of the
senses as leading only to “opinion” not truth, and as
unsuitable for handling the questions that really inter-
ested them. The only appropriate procedure was de-
duction from necessary and indemonstrable first prin-
ciples. With the recovery of the later books of
Aristotle's Organon, the logica nova, men of the thir-
teenth century focussed upon syllogistic logic, and paid
little attention to the problem of method. With a single
important exception, discussion of method was confined
to the medical centers of Italy. And even this re-
markable person was almost certainly influenced by
what he knew of Galen. The exception, of course, is
Robert Grosseteste, to whose role as scientific
methodologist and scientist Professor Crombie has de-
voted a major book. For Grosseteste scientific knowl-
edge is knowledge, as it was for Aristotle, of the causes
of things, knowledge propter quid. The natural proce-
dure in any science is to proceed from those particulars
and whole objects known to us, directly but confusedly
through the senses, up to the principles or causes, and
then by a deductive chain to show the dependence of
the particulars upon the principles or causes. But
Grosseteste's method is primarily dialectical; its aim
is the discovery of a definition, a generalized verbal
characterization, and it is perhaps not surprising that
he discusses composition before taking up resolution,
for in certain sciences (notably mathematics) the syn-
thetic or compositive method is all that seems to be
needed in most cases. A different approach is needed
in physics which is uncertain, because, as Crombie
paraphrases him, there can be only “probable knowl-
edge of changeable natural things” (Crombie, p. 59).
Causal definitions in physics could not be arrived at
a priori (or simpliciter) like the axioms of geometry;
they had to be reached by analysis or resolution of
experimental objects, a process involving first, a dis-
secting of the object or phenomenon, and then, an
inductive leap. But “the special merit” of Grosseteste's
methodology, as Crombie points out, was to recognize
that the induction is not probative, not a demon-
stration. What is necessary is verification or falsification
of the principles or definitions arrived at by analysis
(resolutio). The procedure of deducing the conse-
quences of the definition, cause or principle, is of
course the compositio: it serves to confirm (or falsify)
the analytic results. Together both procedures—res-
and compositio—constitute a single method.
In the study of nature the two procedures must be
used together.

Speculations on method never flourished in Paris or
Grosseteste's Oxford, but attracted much attention in
the universities of northern Italy, in the late fifteenth
century and more especially in the sixteenth century.
Two writers had a particularly great influence,
Agostino Nifo (ca. 1473-1545) and Jacopo Zabarella
(1533-89). Both men, arguing much like Grosseteste,
asserted that the object of a science is to discover the
causes, the propter quid, of observed effects. To dis-
cover the causes, one must first proceed a posteriori,
inferring causes from effects, i.e., using first the method
of resolution or analysis; then the demonstrative or
compositive method can be used to develop the conse-
quences. The double procedure constitutes the method.

Ernst Cassirer was the first to call attention to
Zabarella, to bring him to light once more, and to see
him as an influence on Galileo's scientific method, a
position taken later, and even more strongly, by J. H.
Randall, Jr. (Cassirer, I, 136-44). But Neal Gilbert in
a recent book, Renaissance Concepts of Method (esp.
Ch. 7), casts doubt on this interpretation. As with
Grosseteste, the emphasis of these Renaissance philos-
ophers is on the method for its own sake, on method
as prescriptive for all areas of knowledge; the concern
is not with its application to natural science alone, but
to all disciplines, metaphysical, moral, dialectical. The
empirical element, as Gilbert has pointed out, is weak.
While it is true that in analysis or resolution we pass
from what is better known to what is more remote
from us, the better known “experiences” may not be
observations of scientific fact; they can as well be
“clear and distinct ideas” resulting from the analysis
of thoughts and concepts, and the principles or causes
are verbal definitions. Even if this is somewhat unjust
to Zabarella, there is an important difference between
his method and that of Galileo, and an even greater
gap between Galileo and Newton.

In the probatio duplex—the double method of
Grosseteste and Zabarella—the really probative ele-
ment is supplied by the synthetic, deductive arm. The
analytic or resolutive procedure is merely suggestive
or conjectural. This, it would appear, is also charac-
teristic of Galileo, but with him the empirical element
which Gilbert finds lacking in Zabarella is, of course,
much more important.

In many passages, notably in the Letter to the Grand
Duchess Christina,
Galileo, in phrases that are reminis-
cent of Galen's injunctions, insists that the proper
approach to natural philosophy is to employ jointly
“manifest experiences and necessary proofs”; “direct
experience and necessary demonstrations”; “experi-
ments, long observation, and rigorous demonstration”
(Galileo, trans. S. Drake, pp. 179, 183-84, 186, 197).
In such phrases he seems to be implying a double
method of resolution and composition, of analysis and
synthesis. But how does one carry this out?

The Third Day of the Discorsi throws light on the
matter. Galileo opens with the famous statement of


purpose: that he intends “to set forth a very new
science dealing with a very ancient subject,” the sub-
ject of motion in nature. Concerning this, he remarks,
“I have discovered some properties of it which...
have not hitherto been either observed or demon-
” (Galileo, trans. H. Crew and A. de Salvio, p.
153; emphasis added).

After several pages describing the kinematics of
uniform motion, Galileo enters upon the subject of
accelerated motion:

And first of all it seems desirable to find and explain a
definition best fitting natural phenomena. For anyone may
invent an arbitrary type of motion and discuss its properties
... but we have decided to consider the phenomenon of
bodies falling with an acceleration such as actually occurs
in nature and to make this definition of accelerated motion
exhibit the essential features of observed accelerated mo-

(ibid., p. 160).

This suggests, if his earlier statement about discovering
his results did not satisfy us, that he has observed and
perhaps crudely determined the acceleration of falling
bodies in order to arrive at his definition. Thus at least
a crude analysis of experience led him to the rule
nature might follow, led “by the hand, as it were, in
following the habit and custom of nature herself, in
all her various other processes.” “And this, at last,”
he says in the same paragraph, “... after repeated
efforts we trust we have succeeded in doing. In this
belief we are confirmed mainly by the consideration
that experimental results are seen to agree with and
exactly correspond with those properties which have
been, one after another, demonstrated by us” (ibid.).
Such an experimental confirmation completes the syn-
thesis, or compositional phase, of Galileo's double
method. It is notable that the language used (as later
with Newton) is that of mathematics: kinematic de-
scriptions, measurable and representable (as his pages
show) by numbers and geometry. Galileo's “definitions”
are mathematically symbolized “laws.”

A passage in the Dialogue on the Two Great World
seems to confirm our inference; it contains,
also, an explicit reference to the method of resolution:

Simplicio[:] Aristotle first laid the basis of his argument
a priori, showing the necessity of the inalterability of heaven
by means of natural, evident, and clear principles. He
afterwards supported the same a posteriori, by the senses
and by the traditions of the ancients.
Salviati[:] What you refer to is the method he uses in writ-
ing his doctrine, but I do not believe it to be that with
which he investigated it. Rather, I think it certain that he
first obtained it by means of the senses, experiments, and ob-
servations, to assure himself as much as possible of his con-
clusions. Afterwards he sought means to make them demon
strable. This is what is done for the most part in the
demonstrative sciences; this comes about because when the
conclusion is true, one may by making use of the analytical
methods [methodo resolutivo] hit upon some proposition
which is already demonstrated, or arrive at some axiomatic
principle.... And you may be sure that Pythagoras, long
before he discovered the proof for which he sacrificed a
hecatomb, was sure that the square on the side opposite
the right angle of a right triangle was equal to the squares
on the other two sides. The certainty of a conclusion assists
not a little in the discovery of its proof

trans. S. Drake, pp. 50-51).

What Galileo seems to be saying is that nonrigorous,
exploratory methods, based on trial and experiment,
can lead to a certain degree of probability in the
conclusion. This conclusion can then be demonstrated,
either because (as in the mathematical analysis) it leads
to something already known, or because it leads to
something that can be tested experimentally. A point
worth emphasizing is the stress that Galileo places on
the analytic or resolutive procedure as strongly sug-
gestive, though falling far short of probative demon-
stration. The analytic procedure, as he makes clear,
is the method of discovery (in natural philosophy the
only method of discovery or invention); the synthetic
procedure rounds out the process, and is the method
of final demonstration and formal presentation. In the
seventeenth century the problem of method, as distinct
from logic, became of paramount concern. Indeed—as
the Kneales point out in their Development of Logic
(1962)—this led to a neglect or an impoverishment of
logical studies in this century. This new concern, an
attempt to formulate a doctrine of method in natural
science, scientific method, is first encountered in
Francis Bacon.

Bacon has suffered at the hands of many historians
of science and of philosophy, and he has often been
grossly misinterpreted. His self-appointed role was to
stress experience and experiment, and to do so with
all the rich resources of rhetoric. His aim, as he put
it, is to restore “the commerce between the mind of
men and the nature of things.” In the study of nature
we cannot succeed if we rely excessively or exclusively
on the human reason, “if we arrogantly search for the
sciences in the narrow cells of the human under-
standing, and not submissively in the wider world”
(cited by Basil Willey, p. 36). But any careful reading
of Bacon reveals that his goal is the discovery of axioms
and principles from which a demonstrative science can
be constructed. In any case, these axioms and principles
should not be ad hoc or gratuitous; they should not
be “hypotheses” in Newton's pejorative sense: they
must somehow be rooted in, derived from, Nature
herself. What Bacon wrestles with, if not too success-


fully, is the problem of induction, in other words the
problem of increasing the probative value of the
analytical arm of the double method; since the syn-
thetic arm had been thoroughly investigated from
Aristotle to his own time, it could be momentarily left
aside. To arrive at axioms we must learn how to analyze
and dissect nature, dissecare naturam:

Now what the sciences stand in need of is a form of induc-
tion which shall analyze experience and take it to pieces,
and by a due process of exclusion and rejection lead to an
inevitable conclusion [that is, to axioms and causes]

ed. J. M. Robertson, p. 249).

In his effort to strengthen the upward procedure per-
haps Bacon helped to distract attention away from the
double method. Interest in the double method, and an
appreciation if its power as a scientific instrument,
waned in mid-century. But for this Descartes is perhaps
as much at fault as Bacon; at his hands the “method”
is distorted in a very interesting way. It is Newton who
has the honor of restoring and sharpening it as a tool
of what he called “Experimental Philosophy.”

It has been claimed that the double method of anal-
ysis and synthesis, of resolution and composition, is the
central feature of Descartes' famous method. As every-
one knows, his doctrine of method is set forth in the
readable, but here and there oddly cryptic, Discourse
on Method
(1637). The method is summed up in the
famous four rules which Descartes introduces in Part
II of his book, but more completely set forth in the
twenty-one rules of his posthumous Regulae ad direc-
tionem ingenii
(Oeuvres..., X, 359, 469).

The Discourse, as we commonly encounter it, was
only a preface, an introduction, to those illustrations
of the “method” which were published in the original
book, and which have ever since been generally
omitted from modern editions: the Dioptrique, the
Météores, the Géométrie, intended together to illustrate
the range of application of the method. When one
examines the two physical essays as examples of
Descartes' method, there is certainly no trace of a
double way, a probatio duplex. The Dioptrique begins
with a discussion of the nature of light, and the
phenomena of refraction, presented in synthetic fash-
ion. The Météores is an even better example of hypo-
thetical reasoning, taking its departure from a purely
conjectural picture of the shapes of particles.

It would seem that for Descartes analysis and syn-
thesis are simply two alternative directions in which
one can conduct one's thoughts in orderly fashion.
Analysis, to be sure, is the road to first principles,
leading to the clear and distinct ideas; but it is the
analysis of concepts, of thought, not the analysis of
sense experience or experiments. In either direction
one follows long chains of reasoning in which the
validity of each step involves the spontaneous opera-
tion of the vis cognoscens, the power of the mind to
grasp directly the “simple natures,” the “atoms of
evidence” which are the links of the chain. This power,
or rather the action of the mind at each of these ele-
mentary steps, is what Descartes calls intuitus.

All reasoning, for Descartes, is thus a series of intui-
tive steps. And what men need, instead of the rules
of formal logic which may be dispensed with, are the
practical injunctions of his Four Rules. Method, for
Descartes, is merely order in thought, order that will
permit the natural intellect to operate unimpeded. This
order can be insured by observing the simple rules of
intellectual behavior which Leibniz found so absurdly
obvious yet so vague. Descartes' famous rules are per-
haps best described as propaedeutic, or even as
prophylactic, injunctions. When they are scrupulously
observed, the power of the mind, the vis cognoscens,
operates reliably and surely. There seems to be little
justification for finding in Descartes' method the dual
procedure we have been describing. The truth value
does not come from the mutual support of the two
limbs of a dual method, but from perceiving clear,
distinct, and irrefutable ideas.

Whether or not it is adequate to present Descartes'
“method” in this fashion, one thing at least is certain:
those of his followers who discuss analysis and synthe-
clearly see these as two sorts of method, not as
jointly constituting, when used one after the other, a
single method. Arnauld and Nicolle, the authors of the
book called the Art of Thinking, a work published in
1662 and often called the Port Royal Logic, write as

We distinguish two kinds of method: the one for the dis-
covery of truth is called analysis or the method of resolution
or the method of invention; the second, used to make others
understand the truth, is called synthesis or the method of
or the method of instruction

(Arnauld and
Nicolle, p. 302).

Analysis, they remark farther on, is used “to investigate
a specific thing rather than to investigate more general
things as is done in the method of instruction [i.e.,
composition].” And they add that this analysis “consists
more of discernment and acumen than of particular
procedures,” a statement that reminds us, not only of
Descartes, but of Bacon's remark that analysis by ex-
periment, which he calls the Chase of Pan, is really
a kind of sagacity. And the Port Royal logicians bluntly
state that “the more important of the two methods”
is the method of composition “in that composition is
used for explanation in many disciplines” (ibid., p. 309).

A similar distinction is made by Pierre-Sylvain Régis,


a Cartesian physicist. In his Système de philosophie,
published in 1690, Régis speaks of two methods: “of
which one serves to instruct ourselves and is called
analysis, or the method of division, and the other which
is used to instruct others is called synthesis, or the
method of composition” (cited by Mouy, p. 148).

The same ideas are expressed in W. J. 'sGravesande's
introductio ad philosophiam, metaphysicam et logicam
(Leiden, 1736). Book II is devoted to logic,
and the third part of this is called “On Method.” The
opening words are as follows:

It now remains to indicate the route that the person...
should follow to reach a true understanding of the things
he has set out to examine.
The method should be different, according to the different

First I shall treat the method for discovering truth, and
then the method that we use to explain to others that which
we know.

The first method is called analytic, or the method of
resolution; the other is synthesis or the method of composi-

The general difference between the two methods consists
in this: that in the first method one passes from the complex
to the simple by resolution; and in the second one goes
from the simple to the compounded

(trans. from French
version, ed. J. N. S. Allamand, Part II, p. 120).

We see how different from Newton's these state-
ments are, which is curious when we recall that
'sGravesande is chiefly remembered for his exposition
of the Newtonian philosophy. In his pages on method
the Dutch scientist owes much, it would seem, to the
later Cartesians and perhaps more immediately to the
Port Royal Logic.


Before trying to assess Newton's method of analysis
and synthesis, comparing it with the twofold scheme
so long and so variously elaborated by his predecessors,
it might be well to consider a longer and more relaxed
exposition that Newton never published, and which is
closely related to the famous methodological section
of Query 23/31 cited at the beginning of this article:

As Mathematicians have two Methods of doing things wch
they call Composition & Resolution & in all difficulties have
recourse to their method of resolution before they com-
pound so in explaining the Phaemoena of nature the like
methods are to be used & he that expects success must
resolve before he compounds. For the explications of
Phaenomena are Problems much harder then [sic] those in
Mathematicks. The method of Resolution consists in trying
experiments & considering all the Phaenomena of nature
relating to the subject in hand & drawing conclusions from
them & examining the truth of those conclusions by new
experiments & new conclusions (if it may be) from those
experiments & so proceeding alternately from experiments
to conclusions & from conclusions to experiments untill you
come to the general properties of things, [& by experiments
& phaenomena have established the truth of those proper-
ties.] Then assuming those properties as Principles of Phi-
losophy you may by them explain the causes of such
Phaenomena as follow from them: wch is the method of
Composition. But if without deriving the properties of
things from Phaenomena you feign Hypotheses & think by
them to explain all nature you may make a plausible systeme
of Philosophy for getting your self a name, but your systeme
will be little better than a Romance. To explain all nature
is too difficult a task for any one man or even for any one
age. Tis much better to do a little with certainty & leave
the rest for others that come after you then [sic] to explain
all things by conjecture without making sure of any thing

(Cambridge University Library, MS. Add. 3970 [5]).

Others before Newton used the word “romance” to
describe fanciful hypotheses. See, for example, Henry
Power in his Experimental Philosophy (1664), who
speaks on p. 186 of those “that daily stuff our Libraries
with their Philosophical Romances.”

In contrast to Descartes, the logicians of Port Royal
and 'sGravesande, Newton sees the two methods as
constituting a single procedure, in which one begins
by analysis or resolution, and follows this by a synthetic
demonstration. Formally, this is the double way, the
probatio duplex, of Grosseteste, Nifo, Zabarella, and
the other early methodologists. Unlike them, however,
Newton—like Galileo—would have us analyze not so
much our ideas about things as the phenomena. But
in turn Newton's double method differs from that of
Galileo in a subtle but important way. With Galileo,
as we saw, the analysis by experiment and observations
is merely suggestive or indicative. The real cogency
of the method depends on the demonstration: on syn-
thesis or mathematical deduction. With Newton, how-
ever, the stress is on the analysis which “consists,” as
he says in the Opticks, “in making experiments and
observations and in drawing general Conclusions from
them by Induction.” For Newton the analytic proce-
dure is independently probative, although falling short
of strict demonstration. Indeed (like Bacon before him)
he feels it necessary to stress this analytic procedure,
as he does in the Opticks by devoting more space to
it than to the synthetic arm. Like Descartes and the
Port Royal Logicians, he too sees analysis as the true
method of discovery, of “invention.” We must, he
wrote, admit of “no Objections against the Conclusions,
but such as are taken from Experiments, or other cer-
tain Truths. For hypotheses are not to be regarded in
Experimental Philosophy.” Although this experimental
and inductive process does not lead to demonstration,
“yet it is the best way of arguing which the Nature
of Things admits of” (Opticks, 4th ed. [1730], p. 404).


If the force of Newton's dual method does not wholly
depend (as it seems to have mainly done with Galileo)
upon the synthetic procedure, what does this deductive
limb of the double method actually contribute?
Newton does not restrict it to a confirmatory role; still
less does he limit its use to presenting or teaching what
has already been discovered. The deductive limb can
also be a means of prediction and discovery for, as he
points out in the draft from which we have quoted
above, one can deduce unexpected consequences.
Having discovered “from Phaenomena” the inverse
square law of universal gravitational force, and then
using this force as a Principle of Philosophy, he writes
(in the same unpublished MS cited above):

I derived from it all the motions of the heavenly bodies
& the flux & reflux of the sea, shewing by mathematical
demonstrations that this force alone was sufficient to pro-
duce all those Phaenomena, & deriving from it (a priori)
some new motions wch Astronomers had not then observed
but since appeare to be true, as that Saturn & Jupiter draw
one another, that the variation of the Moon is bigger in
winter then in summer, that there is an equation of the
Moon's meane motion amounting to almost 5 minutes wch
depends upon the position of her Apoge to the Sun.

The later history of science has again and again con-
firmed the power of a well-founded theory to predict
new phenomena, and to explicate other facts which
had not been considered when the theory was

In Query 31, immediately after describing his
method, Newton tells us how the two procedures are
exemplified in the foregoing books of the Opticks (op.
cit., p. 405). In the greater part of the First Book,
Newton sets forth his classic experiments showing that
light is a heterogeneous mixture of rays of different
refrangibility, and that rays of different refrangibility
differ also in color. Although in this book Newton
affects a kind of axiomatic presentation beginning with
definitions and axioms, and enunciating a series of
propositions, the procedures are really analytic in his
sense, as he tells us they are: the propositions are not
abstract mathematical statements, but affirmations of
physical or experimental fact, and they are justified,
not by mathematical deduction, but by what he calls
“proofs by experiment.” These discoveries being
proved, Newton writes, “they may be assumed in the
Method of Composition for explaining the Phaenomena
arising from them.” An example “of which Method I
gave at the end of the first Book.” He does not specify
what propositions he means, but it is clear that he is
referring us to those propositions he designates as
“problems” rather than as “theorems” and which we
encounter in Book I, Part II (ibid., pp. 161-85): to
explain the colors produced by a prism (Prop. VIII.
Prob. III); to elucidate the colors of the rainbow; (Prop.
IX. Prob. IV); and to explain the permanent colors of
natural bodies (Prop. X. Prob. V).


At first sight it may seem odd to find Newton equat-
ing the analytic procedure with experimentation. Yet
if we think about it for a moment, Newton's reasons
are quite clear. A convincing and well-designed exper-
iment involves a sort of dissection of analysis of nature,
an isolation of the phenomenon to be examined, and
the elimination of disturbing factors. As Lavoisier
wrote long after Newton's time:

One of the principles one should never lose sight of in the
art of conducting experiments is to simplify them as much
as possible, and to exclude (écarter) from them all the cir-
cumstances that can complicate their results

[1789], p. 57).

Experiment, indeed, is usually necessary to determine
which factors can safely be eliminated or at least must
be held constant, and which are those that primarily
determine the phenomenon. As W. Stanley Jevons
wrote: “The great method of experiment consists in
removing one at a time, each of those conditions which
may be imagined to have an influence on the result”
(Jevons [1905], p. 417). Physical nature does not readily
reveal its secrets to the phenomenologist, but only to
those who analyze.

Newton, in any event, profoundly altered that con-
ception of experiment which Bacon had advocated and
which in his spirit was accepted by so many virtuosi
of the early Royal Society. Newton's Experimental
Philosophy is not what Thomas Sprat or Henry Power,
or even Robert Boyle, called by that name. Newton
would not have agreed that experiment merely serves
to render plausible the great sweeping “hypotheses”
of the mechanical philosophers. Nor, at the other ex-
treme, could he have agreed with Samuel Parker that
probably “we must at last rest satisfied with true and
exact Histories of Nature” (cited by Van Leeuwen), or
with Locke who argued that improving knowledge of
substances by “experiences and history... is all that
the weakness of our faculties in this state of mediocrity
which we are in this world can attain to” (Essay
Concerning Human Understanding,
Book IV, Chs. 12,
10; cf. Chs. 3, 29).

On the title page of his Experimental Philosophy
(1664), a miscellany of microscopic observations and
experiments with the Torricellian Tube and with the
magnet, Henry Power described them as providing
“some Deductions, and Probable Hypotheses... in
Avouchment and Illustration of the now famous Atom-
ical Hypothesis.” With greater experimental gifts and


a richer scientific imagination, Robert Boyle can be
said to have guided his own investigation in the same

For Newton, experiment is essentially a device for
problem solving, for determining with precision the
properties of things, and rising from these carefully
observed “effects” to the “causes.” More clearly than
Bacon was able to do, Newton showed by his method
that experimentation could lead with at least “moral
certainty” to axioms, principles, or laws. In two ways
Newton's method must be distinguished from that of
the majority of his predecessors and nearly all his
contemporaries. He insisted upon the cogency of a
single, well-contrived experiment to answer a specific
question, as opposed to the Baconian procedure of
collecting and comparing innumerable “instances” of
a phenomenon. Perhaps even more significant,
Newton's experiments, whenever it is possible, are

Robert Hooke, to be sure, was fully capable of
designing and carrying out experiments to test a con-
jecture or working hypothesis. This he did in his “Noble
Experiment” in which, by dissecting away the dia-
phragm of a dog and blowing air through the immobi-
lized lungs, he showed that the animal could be kept
alive, and in this way verified “my own Hypothesis
of this Matter,” namely that it is the air passing into
the blood, not the motion of the lungs, that was neces-
sary for life. Yet in his dispute with Newton over the
latter's first paper on light and color Hooke's arguments
are often Baconian. He argues that Newton's famous
prismatic experiment, what Newton called his experi-
mentum crucis,
being a single isolated experiment, is
unpersuasive, compared to “all the experiments and
observation,” and the “many hundreds of trials” he
(Hooke) had made (Newton, Papers and Letters, pp.

But it was upon this lone experiment, Newton
replied, that “I chose to lay the whole stress of my
discourse.” By the Baconian term experimentum
—a phrase he borrowed from Hooke's Micro-
(1665)—Newton means an experiment de-
signed to decide between two alternative outcomes,
in other words an experiment designed (like Hooke's
“Noble Experiment”) to answer a clearly formulated
question posed to Nature. In adopting this point of
view, Newton of course had distinguished forerunners
in Galileo, William Gilbert, William Harvey, and
Blaise Pascal, among others. But it is interesting to cite
some words of Isaac Barrow, a man to whom Newton
was greatly indebted:

The Truth of Principles [Barrow wrote] does not solely
depend on Induction, or a perpetual Observation of Partic-
ulars, as Aristotle seems to have thought; since only one
Experiment will suffice (provided it be sufficiently clear and
indubitable) to establish a true Hypothesis, to form a true
Definition; and consequently to constitute true Principles.
I own the Perfection of Sense is in some Measure required
to establish the Truth of Hypotheses, but the Universality
or Frequency of Observation is not so

(Barrow, p. 116).


For his notion that scientific investigation should
consist in the solving of discrete, well-defined problems
Newton surely owed much, as the name of Isaac
Barrow suggests, to the mathematical tradition, for that
is how mathematicians of necessity proceed. We should
remember that among students of physical nature, were
men like Galileo, Torricelli, and Pascal—all of them
mathematicians more than natural philosophers—who
pointed the way and demonstrated by their achieve-
ments that this modest, piecemeal approach was the
most fruitful way of studying not only mathematical
problems but also nature. Few if any of these men
would have described what they were doing as
“physics,” for in the seventeenth century “physics”
meant natural philosophy, which was sharply set apart
from mathematics, as it had been since the time of
Aristotle. Subjects like optics, mechanics, music
(acoustics and harmonics), which we now consider
branches of physics, were described as belonging to
the “mixed or concrete mathematics” (ibid., pp. 16-20;
see also Proclus' Commentary on Euclid's Elements,
in Cohen and Drabkin, pp. 2-5). They were subjects
that treated mathematically things perceived by the
senses, whereas pure mathematics dealt only with
things “conceived by the mind.” But they were not
parts of physics (Cohen and Drabkin, pp. 90-91).

Physics in Newton's day was exemplified by those
all-embracing, all-encompassing systems of nature
devised by the so-called mechanical philosophers:
Pierre Gassendi, Thomas Hobbes, Descartes, and their
lesser followers. These men all shared the view—in
opposition to Aristotle with his “substantial forms” and
“occult qualities” and to Paracelsus with his spiritual
agencies—that the underlying principles of physical
nature were to be found in matter and its motions,
and they built their different systems on an all-
embracing mechanism. Common to all their systems,
despite their rejection of Aristotle, was the conviction
that the purpose of physics—a science of nature in a
sense that is almost Aristotle's—was to explain the
visible world in terms of particulate matter: the sizes,
shapes, motions, and mechanical interaction of invisible
particles, or what Francis Bacon had called “the secret
motions of things.” Physics; a branch of philosophy,
was a dialectical science that imparted knowledge,
derived from “first principles,” about the whole mate-


rial universe. As Descartes put it, physics was that
second branch of philosophy (the first being meta-
physics) “in which, after having found the true princi-
ples of material things, one examines in general how
the whole universe is composed; then in particular
what is the nature of this earth and of all the bodies
that are commonly found around her, like air, water,
fire, the magnet and other minerals” (Descartes, IX,
14). Even more self-confident and succinct is the
definition of Descartes' disciple, Jacques Rohault.
Physics, he wrote in his Traité de physique (1671), is
the science “that teaches us the reasons and causes of
all the effects that nature produces” (p. 1).

The logical model for the builders of these systems
was mathematics, and their method of presentation
was, in general, synthetic and deductive. Yet the lan-
guage and syntax are verbal, not mathematical.
Mathematizable in principle, Descartes' Principles of
has no trace of mathematics. Indeed be-
cause of this widely accepted separation between the
disciplines of mathematics and physics, a mathematical
physics appeared to most men to be a contradiction
in terms. But there are important exceptions.

One of the earliest is Galileo, who wrote in his
Saggiatore a passage that has often been quoted:

Philosophy is written in that great book which ever lies
before our eyes—I mean the universe—but we cannot un-
derstand it if we do not first learn the language, and grasp
the symbols, in which it is written. This book is written
in the mathematical language, and the symbols are triangles,
circles, and other geometrical figures, without whose help
it is impossible to comprehend a single word of it; without
which one wanders in vain through a dark labyrinth

(Galileo, V, 6, p. 232).

Even more eloquent in opposing the conventional split
between mathematics and natural philosophy was Isaac
Barrow, one of that small number of Englishmen who
had mastered Galileo's work, and from whom, in all
likelihood, Newton was led to understand the thought
and achievement of the great Italian scientist. Barrow,
in discussing those “Sciences termed Mixed Mathe-
” commented:

I suppose they ought all to be taken as Parts of Natural
being the same in Number with the Branches of
Physics.... For these mixed Sciences are stiled Mathe-
matical for no other Reason, but because the Consideration
of Quantity intervenes with them, and because they require
Conclusions to be demonstrated in Geometry, applying
them to their own particular Matter. And, according to the
same Reason, there is no Branch of natural Science that
may not arrogate the Title to itself; since there is really
none, from which the Consideration of Quantity is wholly
excluded, and consequently to which some Light or Assist-
ance may not be fetched from Geometry.

And he goes on, in what is almost a paraphrase of
the famous Galilean passage:

For Magnitude is the common Affection of all physical
Things, it is interwoven in the Nature of Bodies, blended
with all corporeal Accidents, and well nigh bears the prin-
cipal Part in the Production of every natural Effect

(Barrow, p. 21).

Elsewhere Barrow wrote that no one can expect to
understand or unlock the hidden meanings of nature
without the “Help of a Mathematical Key”:

For who can play well on Aristotle's Instrument but with
a Mathematical Quill; or not be altogether deaf to the
Lessons of Natural Philosophy, while ignorant of Geometry?

(ibid., pp. xxvi-xxvii).

We need hardly stress the essentially mathematical
character of Newton's major work, The Mathematical
Principles of Natural Philosophy.
If a glance at the book
were not enough to convince us, Newton makes sure
that we understand what he is about, and how he has
bridged the gulf between mathematics and physics. In
his Preface he writes that like “the moderns” he has
in his treatise “cultivated mathematics as far as it
relates to philosophy” and offers his book... as the
mathematical principles of philosophy, for the whole
burden of philosophy seems to consist in this—from
the phenomena of motions to investigate the forces of
nature, and then from these forces to demonstrate the
other phenomena... (Principia, ed. F. Cajori, pp.

By “philosophy” Newton means, of course, natural
philosophy or “physics.” And he seems in this passage
to refer to the traditional distinction between mathe-
matics and physics. Yet he makes clear that these
“principles”—the laws and conditions of certain mo-
tions and of powers or forces—are the things “we may
build our reasonings upon in philosophical inquiries.”
One passes indeed without difficulty from one domain
to the other:

In mathematics we are to investigate the quantities of forces
with their proportions consequent upon any conditions
supposed; then, when we enter into physics, we compare
those proportions with the phenomena of Nature...

p. 192; emphasis added).

One question immediately confronts us: did Newton
conceive of the famous method—his double procedure
of analysis and synthesis set forth and exemplified in
the Opticks, and where the analytic arm is identified
with experiment and observation—as applying equally
well to the Principia? This has recently been denied,
yet the answer is surely in the affirmative. To be sure,
the two works offer a striking contrast; they treat not
only different aspects of nature, but at first glance seem


to treat them in different ways. The Principia is, at
least in the first two books, a work of abstract rational
mechanics, strictly mathematical and presented in
axiomatic fashion. A chain of propositions treat of mass
points or idealized spherical bodies subject to certain
imagined forces. Yet even when the results are applied
to “physics”—to the real bodies of the solar system—
these are treated as bodies qualitatively similar, de-
prived of what John Locke would have called their
“secondary qualities” and differing only in such
quantifiable properties as mass, extension, impene-
trability, and state of rest or motion.

Abstract and mathematical though it appears
throughout, the Principia was deemed by Newton to
be as firmly rooted in observation and experiment as
the Opticks. In the Scholium to the axioms or laws of
motion Newton wrote that “I have laid down such
principles as have been received by mathematicians,
and are confirmed by abundance of experiments” (ibid.,
p. 21). This, he felt, need not be insisted upon for the
first two laws of motion. But his third law, the law
of equality of action and reaction, he saw to be a novel
assertion requiring further justification. To this end he
invoked at some length the experiments on elastic
impact carried out some years before independently
by Christopher Wren, John Wallis, and Christian
Huygens; and he concludes that “so far as it regards
percussions and reflections [the third law] is proved
by a theory exactly agreeing with experience” (ibid.,
p. 25). To show further that the law can be extended
to attractions, he cites an experiment he has made on
the mutual attraction of a lodestone and iron. And
elsewhere throughout the work we find scholia serving
the same purpose of supporting important propositions
by experimental evidence. Many years later, in the
unpublished discussion of his method of analysis and
composition (the first part of which was quoted above)
he wrote:

Thus in the Mathematical Principles of Philosophy I first
shewed from Phaenomena that all bodies endeavoured by
a certain force proportional to their matter to approach
one another, that this force in receding from that body
grows less & less in reciprocal proportion to the square of
the distance from it & that it is equal to gravity & therefore
is one and the same force with gravity

(loc. cit.).

Having tried to persuade us that the famous princi-
ple and law of universal gravitation was discovered
through analysis, he describes in the passage quoted
earlier his subsequent use of the synthetic method.

Newton's Opticks, by contrast, deals with the “sec-
ondary qualities” of things: chiefly color and—if we
take the famous Queries into consideration—those
attributes which differentiate various kinds of bodies:
chemical behavior, phenomena associated with heat,
and such physical properties as cohesion, surface ten-
sion, and capillary rise (I. B. Cohen [1956], pp. 115-17).

Yet it is wrong to insist, as one scholar has done,
that there are two kinds of Newtonianism: the mathe-
matical Newtonianism of the Principia and the “exper-
imental Newtonianism” of the Opticks. To remove any
reasonable doubt as to what Newton himself thought,
we may quote from an anonymous review what are
generally acknowledged to be his own words:

The Philosophy which Mr. Newton in his Principles and
Optiques has pursued is Experimental; and it is not the
Business of Experimental Philosophy to teach the Causes
of things any further than they can be proved by Experi-

(Philosophical Transactions, 19, no. 342 [1717], 222).

The Opticks, unlike the Principia, consists largely of
a meticulous account of experiments. Yet it can hardly
be called nonmathematical, although little more than
some simple geometry and arithmetic is needed to
understand it. In spirit it is as good an example of
Newton's “mathematical way” as the Principia: light
is treated as a mathematical entity, as rays that can
be represented by lines; the axioms with which he
begins are the accepted laws of optics; and numbers—
the different refrangibilities—serve as precise tags to
distinguish the rays of different colors and to compare
their behavior in reflection, refraction, and diffraction.
Wherever appropriate, and this is most of the time,
his language of experimental description is the lan-
guage of number and measure. It is this which gives
Newton's experiments their particular cogency.

Observation is not merely looking and seeing; it is
a kind of reporting. We report to ourselves or to others
some aspect of an object that arouses our interest. In
this broad sense a painting or a poem is a kind of
report. Some aspect of visual or auditory or tactile
experience is singled out from the flux of nature to
be attended to. But not all reports, as we know to our
sorrow, are really observations. Any observation de-
serving the name, certainly any observation we might
call “scientific,” involves a comparison with something
else. And the most precise and unambiguous compari-
sons are those expressed in the language of number
and measure. When we measure we do not simply
contrast two objects with one another. We do not just
report that object A appears bigger, heavier, brighter,
or faster than object B; we report how much they differ
from each other. What is required is some way of
attaching a more precise meaning to “bigger,”
“heavier,” and so on. This we do by comparing both
objects with some standard. Just as in counting we
compare a set of objects with that abstract standard
or scale we call the system of natural numbers, so when


we measure we physically compare the objects at hand
with a unit or standard of measure, which in turn
involves comparing both the object and the standard
with our abstract numerical scale. When we perform
this operation of comparison, using the language of
numbers—that is, when we measure—we are reporting
this relationship of the objects as ratios. This, indeed,
is the meaning that Newton attaches to the word

Newton interprets the numbers themselves as ratios
or measures. Thus he writes: “By Number we under-
stand not so much a Multitude of Unities, as the
abstracted Ratio of any Quantity, to another Quantity
of the same Kind, which we take for Unity. And this
is threefold; integer, fracted, and surd: An Integer is
what is measured by Unity, a Fraction, that which a
submultiple Part of Unity measures, and a Surd, to
which Unity is incommensurable” (Universal Arithme-
London [1728], p. 2).

An experiment is, of course, only a contrived obser-
vation, and all the advantages of precision and lack
of ambiguity that accrue to observations by being cast
in the language of number and measure must neces-
sarily be found in what we call “quantitative experi-
ments,” which Newton's almost always are. There is
no better instance of Newton's quantitative approach
to his experiments than the following undated manu-
script page describing things “To be tryed” to elucidate
the phenomenon of diffraction, that is the bending of
light, and the production of colored fringes (fasciae
to Newton), when light passes through a tiny hole or
past a knife-edge:

  • 1. What are the numbers limits and dimensions of the
    shadow & fasciae of a hair illuminated from a point at
    several distances.

  • 2. How far a hair in the edge of light casts light into the
    shadow surrounding the light.

  • 3. Whether in the approach of a hair to the shaddow the
    fasciae encreas & wch fasciae vanish first.

  • 4. How many fasciae can be seen through a Prism.

  • 5. At what distances each fascia begins to appear.

  • 6. What alteration is made by the bluntness & shapness [sic]
    of the edge or by the density of the matter.

  • 7. How much the shadow of a pin or slender wiar is broader
    then that of a hair.

  • 8. Whether one hair behind another make a broader shadow
    & how much.

  • 9. At what distances from one another two hairs, two backs
    or edges of knives or raisors, two wiars or pins & two larger
    iron cilinders make their fasciae meet.

    How the same or other bodies make their fasciae go into
    one anothers shadows.

    In what order the fasciae begin to appear or disappear
    increase or decreas [sic] in going into or out of any well
    defined shadow (Cambridge University Library, MS. Add.
    3970 fol. 643; emphasis added).

From a series of observations men habitually are
impelled to generalize. To generalize is to report and
sum up in some tidy way the results of a series of
comparisons. The pitfalls of ordinary language com-
pound the dangers of the generalizations we make in
everyday life. But even the murky business of general-
izing—of making an inductive inference—gains pre-
cision through the use of numbers, of mathematical
rather than verbal language. The end product is a
mathematically expressed “rule,” or “law,” or—to
employ a favorite word of Newton's day—a “princi-
ple.” Thus Newton opens the Opticks with what he
called “Axioms” which are simply the well-established
laws of geometrical optics. When, on the other hand,
he enunciates a law that he has himself discovered,
a generalization that he has reached by an inductive
inference and which is quantitatively expressed, he
often employs the word “rule.” Thus after reporting
a series of detailed measurements on the colored rings
produced when light passes through thin, transparent
bodies, he concludes: “And from these Measures, I
seem to gather this Rule: That the thickness of the Air
is proportional to the secant of an angle, whose Sine
is a certain mean proportional between the Sines of
Incidence and Refraction” (Opticks [1704], Book II,
Part I, p. 12).

Clearly Newton's extreme confidence in his Method
of Analysis, in the probative power of inductive in-
ferences from his experiments, depends not a little on
the fact that the ascending chain of comparisons by
which he reaches these “rules” or “laws” is expressed
in the language of mathematics. This use of number,
one hardly needs to add, strengthens the deductive,
synthetic limb of his double method, for the syntax of
mathematical demonstration is at his disposal, in pur-
suing the downward path from “principles” and “laws”
back to the phenomena. It is a syntax well understood
and devoid of the ambiguities and traps of purely
verbal deduction. E. W. Strong, in his “Newton's
Mathematical Way” (in Roots of Scientific Thought,
eds. Wiener and Noland), summed the matter up when
he wrote: “Newton's 'mathematical way' encompasses
both experimental investigation and demonstration
from principles, that is, from laws or theorems estab-
lished through investigation” (p. 413), and this pro-
cedure “requires measures for the formulation of prin-
ciples in optics and mechanics—principles that
incorporate a rule of measure. Were there not mathe-
matical determinations in the experiment, there would
be no subsequent determination in the demonstration”
(p. 421).



Aristotle, Physics, Book I; Nicomachean Ethics, Book III;
cf. Thomas L. Health, Mathematics in Aristotle (Oxford,
1949), pp. 270-72. F. Bacon, The Philosophical Works of
Francis Bacon,
ed. J. M. Robertson (London, 1905), p. 249.
Isaac Barrow, Mathematical Lectures read in the Publick
Schools at the University of Cambridge,
trans. John Kirkby
(London, 1734); for Barrow's familiarity with Galileo's works
see Marie Boas Hall, “Galileo's Influence on Seventeenth-
Century English Scientists,” in Galileo, Man of Science, ed.
Ernan McMullin (New York and London, 1967), pp. 411-12.
Ernst Cassirer, Das Erkenntnisproblem, 3 vols. (Berlin,
1922-23), I, 136-44; for Randall's view, see his well-known
paper “Scientific Method in the School of Padua,” Journal
of the History of Ideas,
1 (1940), 177-206, and its revision
in his The School of Padua and the Emergence of Modern
(Padua, 1961). Cassirer accepted the results of
Randall's inquiry but could not “subscribe to his conclu-
sions,” for he believed Galileo's conception of the dual
method, despite the identity of the terms used, to be more
influenced by the mathematical tradition than by the phi-
losophers of Padua; see his “Galileo's Platonism,” in M. P.
Ashley Montagu, ed. Studies and Essays in the History of
Science and Learning
(New York, 1946), pp. 279-97. I. B.
Cohen, Franklin and Newton, An Inquiry into Speculative
Newtonian Experimental Science
... (Philadelphia, 1956).
M. R. Cohen and I. Drabkin, eds., A Source Book in
Greek Science
(New York, 1948; Cambridge, Mass. 1959). A.
Crombie, Robert Grosseteste and the Origins of Experimental
(Oxford, 1953). Descartes, Oeuvres de Descartes, eds.
C. Adam and P. Tannery, 13 vols. (Paris, 1891-1912). Galen,
Claudii Galeni Opera omnia, ed. C. G. Kühn, 20 vols.
(Leipzig, 1821-33); idem, Galen on Medical Experience, ed.
and trans. R. Walzer (London and New York, 1944). Galileo
Galilei, “Letter to the Grand Duchess Christina” (1615),
in S. Drake, Discoveries and Opinions of Galileo (Garden
City, N.Y., 1957); idem, Opere di Galileo Galilei, ed. A.
Favaro, 20 vols. (Florence, 1890-1909; reprint 1929-39);
idem, Dialogues Concerning Two New Sciences, trans. H.
Crew and A. de Salvio (New York, 1914); idem, Dialogue
Concerning the Two Chief World Systems,
trans. S. Drake
(Berkeley and Los Angeles, 1953). Neal Gilbert, Renaissance
Concepts of Method
(New York and London, 1960). Thomas
L. Hankins, Jean D'Alembert—Science and the Enlighten-
(Oxford, 1970), passim. Jaako Hintikka, “Kant and the
Tradition of Analysis,” Deskription, Analytizität und Ex-
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schungszentrum für Grundfragen der Wissenschaften Salz-
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but in one place modified it to read experimentum crucis.
See Richard S. Westfall, “The Development of Newton's
Theory of Color,” Isis, 53 (1962), 354, and note 46. For a
skeptical appraisal of Newton's famous experiment see
A. I. Sabra, Theories of Light from Descartes to Newton
(London, 1967), pp. 294-97. Sabra's argument that only
Newton's adherence to a corpuscular doctrine permitted
him to infer the heterogeneity of white light from this
experiment is inconclusive. W. S. Jevons, The Principles of
(London and New York, 1905). P. S. de Laplace,
Exposition du système du monde, 6th ed. (Paris, 1835).
A. L. Lavoisier, Traité élémentaire de chimie (Paris, 1789).
Colin Maclaurin, Account of Sir Isaac Newton's Philo-
sophical Discoveries,
3rd ed. (London, 1775). Paul Mouy,
Le Développement de la physique cartésienne, 1646-1712
(Paris, 1934). Isaac Newton, Mathematical Principles of
Natural Philosophy,
ed. F. Cajori (Berkeley, 1934); idem,
Opticks, 4th ed. (1730); idem, Isaac Newton's Papers and
Letters on Natural Philosophy,
ed. I. B. Cohen (Cambridge,
Mass., 1958); idem, “Account of the Booke entituled Com-
mercium Epistolicum, etc.,” Philosophical Transactions, 19,
no. 342 (1717); trans. “Recensio,” in the second edition
(1722) of the Commercium; for Newton's authorship of this
“Account” see Louis T. More, Isaac Newton (New York,
1934), pp. 590-91, note 43; idem, Universal Arithmetick
(London, 1728). Jacques Rohault, Traité de physique (Paris,
1671). W. J. 'sGravesande, Introductio ad philosophiam,
metaphysicam et logicam continens
(Leiden, 1736); trans.
into French as Oeuvres philosophiques et mathématiques de
Mr G. J. 'sGravesande,
ed. J. N. S. Allamand, two parts in
one vol. (Amsterdam, 1774). E. W. Strong, “Newton's
'Mathematical Way'” in Roots of Scientific Thought, eds.
Philip P. Wiener and Aaron Noland (New York, 1957); the
article is reprinted, somewhat abridged, from the Journal
of the History of Ideas,
12 (1951), 90-110. Henry G. Van
Leeuwen, The Problem of Certainty in English Thought
(The Hague, 1963). Basil Willey, The Seventeenth Century
(London, 1949).


[See also Baconianism; Classification of the Sciences;
Cosmology; Experimental Science; Newton's Opticks;
Number; Optics; Unity of Science.]