カバリエリの原理が理解できないというKids。３次元ではなくて２次元で考えるとわかりやすいのでは？
色々と歴史を調べなきゃならないね。
The concept of an indivisible
plato.stanford.edu/entries/continuity/
The concept of an indivisible is closely allied to, but to be distinguished from, that of an infinitesimal.
An indivisible is, by definition, something that cannot be divided, which is usually understood to mean that it has no proper parts.
Now a partless, or indivisible entity does not necessarily have to be infinitesimal: souls, individual consciousnesses, and Leibnizian monads all supposedly lack parts but are surely not infinitesimal. But these have in common the feature of being unextended; extended entities such as lines, surfaces, and volumes prove a much richer source of “indivisibles”.
Indeed, if the process of dividing such entities were to terminate, as the atomists maintained, it would necessarily issue in indivisibles of a qualitatively different nature. In the case of a straight line, such indivisibles would, plausibly, be points; in the case of a circle, straight lines; and in the case of a cylinder divided by sections parallel to its base, circles.
In each case the indivisible in question is infinitesimal in the sense of possessing one fewer dimension than the figure from which it is generated. In the 16th and 17th centuries indivisibles in this sense were used in the calculation of areas and volumes of curvilinear figures, a surface or volume being thought of as a collection, or sum, of linear, or planar indivisibles respectively.
The concept of infinitesimal was beset by controversy from its beginnings.
The idea makes an early appearance in the mathematics of the Greek atomist philosopher Democritus c. 450 B.C.E., only to be banished c. 350 B.C.E. by Eudoxus in what was to become official “Euclidean” mathematics.
We have noted their reappearance as indivisibles in the sixteenth and seventeenth centuries: in this form they were systematically employed by Kepler, Galileo's student Cavalieri, the Bernoulli clan, and a number of other mathematicians.
In the guise of the beguilingly named “linelets” and “timelets”, infinitesimals played an essential role in Barrow's “method for finding tangents by calculation”, which appears in his Lectiones Geometricae of 1670. As “evanescent quantities” infinitesimals were instrumental (although later abandoned) in Newton's development of the calculus, and, as “inassignable quantities”, in Leibniz's.
The Marquis de l'Hôpital, who in 1696 published the first treatise on the differential calculus (entitled Analyse des Infiniments Petits pour l'Intelligence des Lignes Courbes), invokes the concept in postulating that
“a curved line may be regarded as being made up of infinitely small straight line segments,”
and that
“one can take as equal two quantities differing by an infinitely small quantity.”
We are all familiar with the idea of continuity. To be continuous is to constitute an unbroken or uninterrupted whole, like the ocean or the sky. A continuous entity—a continuum—has no “gaps”.
Opposed to continuity is discreteness: to be discrete is to be separated, like the scattered pebbles on a beach or the leaves on a tree. Continuity connotes unity; discreteness, plurality.
While it is the fundamental nature of a continuum to be undivided, it is nevertheless generally (although not invariably) held that any continuum admits of repeated or successive division without limit.
This means that the process of dividing it into ever smaller parts will never terminate in an indivisible or an atom—that is, a part which, lacking proper parts itself, cannot be further divided. In a word, continua are divisible without limit or infinitely divisible.
The unity of a continuum thus conceals a potentially infinite plurality.
In antiquity this claim met with the objection that, were one to carry out completely—if only in imagination—the process of dividing an extended magnitude, such as a continuous line, then the magnitude would be reduced to a multitude of atoms—in this case, extensionless points—or even, possibly, to nothing at all.
But then, it was held, no matter how many such points there may be—even if infinitely many—they cannot be “reassembled” to form the original magnitude, for surely a sum of extensionless elements still lacks extension.
Moreover, if indeed (as seems unavoidable) infinitely many points remain after the division, then, following Zeno, the magnitude may be taken to be a (finite) motion, leading to the seemingly absurd conclusion that infinitely many points can be “touched” in a finite time.
Such difficulties attended the birth, in the 5th century B.C.E., of the school of atomism. The founders of this school, Leucippus and Democritus, claimed that matter, and, more generally, extension, is not infinitely divisible. Not only would the successive division of matter ultimately terminate in atoms, that is, in discrete particles incapable of being further divided, but matter had in actuality to be conceived as being compounded from such atoms.
In attacking infinite divisibility the atomists were at the same time mounting a claim that the continuous is ultimately reducible to the discrete, whether it be at the physical, theoretical, or perceptual level.
The eventual triumph of the atomic theory in physics and chemistry in the 19th century paved the way for the idea of “atomism”, as applying to matter, at least, to become widely familiar: it might well be said, to adapt Sir William Harcourt's famous observation in respect of the socialists of his day,
“We are all atomists now.”
Nevertheless, only a minority of philosophers of the past espoused atomism at a metaphysical level, a fact which may explain why the analogous doctrine upholding continuity lacks a familiar name: that which is unconsciously acknowledged requires no name.
Peirce coined the term synechism (from Greek syneche, “continuous”) for his own philosophy—a philosophy permeated by the idea of continuity in its sense of “being connected”. In this article I shall appropriate Peirce's term and use it in a sense shorn of its Peircean overtones, simply as a contrary to atomism.
I shall also use the term “divisionism” for the more specific doctrine that continua are infinitely divisible.
eigotown.com/eigocollege/marie_english/backnumber/marie_english06
アメリカの公立小中学校や幼稚園では、毎朝、生徒がアメリカの国旗、星条旗に向かって右手を左胸にあて、「忠誠の誓い（the Pledge of Allegiance）」という文句を唱えています。
どんな文句かご紹介しましょう。
I pledge allegiance to the flag of the United States of America and to the Republic for which it stands, one Nation under God, indivisible, with liberty and justice for all.
「私はアメリカ合衆国の国旗と、その国旗が象徴する共和国、神のもとに統一さ れ全ての人々に自由と正義が約束された不可分の国に忠誠を誓います」
幼稚園の子どもたちや小学校低学年の生徒は、indivisibleという単語の意味も分からずに唱えているわけですが、毎日言っている間に自然と国旗を敬う心やアメ リカへの忠誠心が身に付いてしまう、というわけです。
移民の国アメリカは、
「常に愛国心を煽っておかないと国家としての統一が取れ なくなってしまうかも」
という懸念がどうしてもぬぐえないので、幼いうちから「忠誠の誓い」で「洗脳」してる、という見方もありますが、９割方のアメリカ人はこのしきたりを「愛国心形成の礎」とし
て好意的にとらえています。
2002年６月に、第九巡回裁判所*が、
「この誓いの『神のもとに （under God）』という一言は、政教分離を定めた合衆国憲法修正第一条に違反している」
というカリフォルニア州の無神論者（atheist）* Michael Newdow 氏の訴えを認めたときは、アメリカ中がこの判決を非難しました。
その後、同裁判所がこの判決を無効としたため、Newdow 氏は上訴して、今最高裁で審議されている最中です。
アメリカ人の８割以上が、「神のもとに」（under God）を「忠誠の誓い」の中に残すことに賛成していますが、その理由は大きく分けると
1 アメリカの伝統だから
2 アメリカはキリスト教の信念に基づいて建国された国だからという二つ。
2 アメリカはキリスト教の信念に基づいて建国された国だからという二つ。
「忠誠の誓い」 の大もと は、社会主義者（socialist）の作家、フランシス・ベラミー（Francis Bellamy：1855 - 1931)が作った誓いの言葉
“I pledge allegiance to my Flag and the Republic for which it stands, one nation indivisible, with liberty and justice for all.”
で、1892年に子ども向けの雑誌のアメリカ大陸発見400周年記念号に掲載されました。
その後、この誓いの言葉はハリソン大統領（1889-1893年 共和党）のお墨付きを得て公立学校で暗礁されるようになりました。
1924年に「my Flagだと移民が混同するといけない」という心配りから
I pledge allegiance to the flag of the United States of America and to the Republic for which it stands, one Nation, indivisible, with liberty and justice for all.
と改訂されました。
で、問題の「神のもとに」（under God）が付け足されたのは、冷戦の真っ最中の1954年、共和党上院議員ジョセフ・マッカーシー（Joseph McCarthy）が煽動した赤狩り*がアメリカを席巻した直後のこと。
非常に保守的なカトリック教会の組織「コロンブス騎士会」*が、「無神論の共産主義国ソ連とアメリカをしっかり区別するために」と働きかけて、「神のもとに」（under Go
d）という一言が付加されることになったのです。
つまり、「忠誠の誓い」の歴史は、under Godなしのバージョンが62年、ありのバージョンは50年なので、なしのバージョンのほうが歴史的価値は高い、というわけなんです。
だから、「伝統」を重視するのであればベラミーが書いたオリジナルに近い、under Godなしのバージョンを採用すべきなんですよね。
「神のもとに」（under God）を残すことに賛成している理由の２に関しては、「そう思ってる人には何を言ってもムダ」、という感があるのでここでは言及を避けますが、最高裁の判決が出たらまたこの話題にふれますので、お楽しみに！