きわめて単純で容易な，推論の長い連鎖は，幾何学者たちがつねづね用いてどんなに難しい証明も達成する。それはわたしに次のことを思い描く機会をあたえてくれた。人間が認識しうるすべてのことがらは，同じやり方でつながり合っている，真でないいかなるものも真として受け入れることなく，一 つのことから他のことを演縛するのに必要な順序をつねに守りさえすれば，どんなに遠く離れたものにも結局は到達できるし，どんなに隠れたものでも発見できる，と。それに，どれから始めるべきかを探すのに，わたしはたいして苦労しなかった。もっとも単純で，もっとも認識しやすいものから始めるべきだと，すでに知っていたからだ。そしてそれまで学問で真理を探究してきたすべての人びとのうちで，何らかの証明（つまり，いくつかの確実で明証的な論拠）を見いだしえたのは数学者だけであったことを考えて，わたしは，これらの数学者が検討したのと同じ問題から始めるべきだと少しも疑わなかった。もっともそこからわたしが期待した効用は，精神が真理に専心し，誤った論拠に満足しないよう習慣づけることだけだつたけれど。しかし，だからといって，ふつう数学と呼ばれている，あの個々の学科すべてを学ぼうとするつもりはなかった。これらの学科が，対象は異なっても，そこに見いだされるさまざまな関係つまり比例だけを考察する点で一致することになるのを見て，こう考えた。これらの比例だけを一般的に検討するのがよい，その際そうした比例を，わたしにいっそう容易に認識させてくれるのに役立つような対象があれば，そのなかにだけ想定し，しかもそうした対象にだけ限るのではなく，それが当てはまるような他のすべての対象にも，後になっていっそううまく適用できるようにする，と。次に，こうした比例を認識するために，ときにはそれらを一つ一つ別々に考察する必要があり，ときにはただそれらを記憶にとどめ，多くを一度に把握する必要があるのに気がついて，こう考えた。比例を個別的にいっそうよく考察するためには，これを線として想定すべきこと。線以上に単純で，線以上に判明にわたしの想像力や感覚に表象できるものはなかったからだ。しかし，それらの比例を記憶に保持し，多くを一度に捉えるためには，できるだけ短い，ある種の記号で示す必要があること。そしてこのようなやり方で，幾何学的解析と代数学とのあらゆる長所を借り，しかも一方の短所すべてをもう一方によって正せる，と。
The long chains of simple and easy reasonings
by means of which geometers are accustomed
to reach the conclusions of their most difficult
demonstrations, had led me to imagine that
all things, to the knowledge of which man
is competent, are mutually connected in the
same way, and that there is nothing so far
removed from us as to be beyond our reach,
or so hidden that we cannot discover it,
provided only we abstain from accepting the
false for the true, and always preserve in
our thoughts the order necessary for the
deduction of one truth from another. And
I had little difficulty in determining the
objects with which it was necessary to commence,
for I was already persuaded that it must
be with the simplest and easiest to know,
and, considering that of all those who have
hitherto sought truth in the Sciences, the
mathematicians alone have been able to find
any demonstrations, that is, any certain
and evident reasons, I did not doubt but
that such must have been the rule of their
investigations. I resolved to commence, therefore,
with the examination of the simplest objects,
not anticipating, however, from this any
other advantage than that to be found in
accustoming my mind to the love and nourishment
of truth, and to a distaste for all such
reasonings as were unsound. But I had no
intention on that account of attempting to
master all the particular Sciences commonly
denominated Mathematics: but observing that,
however different their objects, they all
agree in considering only the various relations
or proportions subsisting among those objects,
I thought it best for my purpose to consider
these proportions in the most general form
possible, without referring them to any objects
in particular, except such as would most
facilitate the knowledge of them, and without
by any means restricting them to these, that
afterwards I might thus be the better able
to apply them to every other class of objects
to which they are legitimately applicable.
Perceiving further, that in order to understand
these relations I should sometimes have to
consider them one by one, and sometimes only
to bear them in mind, or embrace them in
the aggregate, I thought that, in order the
better to consider them individually, I should
view them as subsisting between straight
lines, than which I could find no objects
more simple, or capable of being more distinctly
represented to my imagination and senses;
and on the other hand, that in order to retain
them in the memory, or embrace an aggregate
of many, I should express them by certain
characters the briefest possible. In this
way I believed that I could borrow all that
was best both in Geometrical Analysis and
in Algebra, and correct all the defects of
the one by help of the other.