新式算術講義

• 底本: 高木貞治 著『新式算術講義』,博文館,明37.7. 国立国会図書館デジタルコレクション:info:ndljp/pid/827403

緖言

普通敎育の程度を越えて，初等數學を修むる人の參考に適せる書籍の，本邦出版界に殆ど絕無なるは，著者の私に憾とする所なり．然れども初等數學の範圍は廣大にして，其分科は繁多なり．今其全般を通じ，細目に入りて，普く此缺陷を塡充せんこと，短日月を以て成さるべき事業に非ず．此小册子は主として材料を算術の範圍內より採り，其最重要なる問題を選み，數學最新の發達によりて占め得たる立脚點より之を觀察して，成るべく簡明なる解釋を試む．

普通敎育に於ける算術の論ずる所は一見甚卑近なるが如しと雖も，若し深く問題の根柢に穿入せんとするときは，必しも然らず．夫れ敎師は其敎ふる所の學科につきて含蓄ある知識を要す．算術敎師が算術の知識を求むる範圍，其敎ふる兒童の敎科用書と同一程度の者に限らるゝこと，極めて危殆なりと謂ふべし．確實なる知識の缺乏を補ふに，敎授法の經驗を以てせんとするは，「無き袖を振はん」とするなり，是を以て此書は廣く算術の敎授に從事する敎師諸氏の中に其讀者を求めんと欲す．

又數學を專攻せんとする學生にありても，目下の狀態に於ては，其算術の知識は幼時普通敎育によりて得たる所に限られ，漸く進んで稍〻高等なる數學諸分科の修業に入るに當りても，數學の根源に關せる問題を囘顧して，精密に之を復習するの遑なきが如し．斯の如くなれば，其知識は堅牢なる地盤を缺くが故に，學ぶ所愈〻進むに隨ひ，知る所愈〻不確實となる．是寔に憂ふべし．偶〻此缺點を覺りて自ら之を補充せんとする者ありとも，恰當なる參考書の缺如せるが爲に感ずる不便決して少小ならず．抑〻算術は汎く數に關する根本的の觀念を論ず，是故に其範圍意外に廣大にして，若し汎く諸分科の專門的書籍を涉獵して其知識を拾集せんと欲せば，少とも整數論，代數學及函數論の一班を窺はざるべからず．此書は斯の如き摸索の勞を節せしめんが爲に，最重要なる問題を集結して，之に最近接し易き解說を與へんことを期す．

十一章七十五節より成れる此書の內容は，自ら分れて二部となる．其前一半は第一章より第七章に至り，專ら有理數を論ず．これ比較的最近よく世に知られたる事實に關せるが故に，敍述の方法は成るべく新奇なるを選み，以て多數の讀者の熟知せる所の者を徒に反復するを避けんとす．蓋し同一の事實を多樣の見地より觀察するは，卽其知識を確實ならしむる所以なり．第四章及第六章に於て，整數論に關する事項の爲に，比較的多大の頁數を割けること，稍〻權衡を失するの觀なきにしもあらずと雖，是一は最少く普通に知られたる所に最多くの力を致さんとする趣旨に出で，又一は數を觀察するに當り，其大小に關せる側面に偏して，數の個性（アリスメチカル，キヤラクター）を藐視すること，決して數の知識を精確ならしむる所以にあらざるを信ぜるに由れり．

第八章以下は袖象的の量として數を論ず．其目的，數とは何ぞ，量を計るとは何の謂ぞ，との卑近なる問題を解釋するにあり．數の觀念を闡明して，數學に牢然動すべからざる基礎を與へたること，實に十九世紀に於ける數學進步の異彩にして又其根源なり．斯の如き高等數學の進步は決して初等數學に影響する所なくして已むべからず．高等數學の論ずる所は槪して通俗の說明に適せずと雖，凡そ極めて根本的なる問題は，之を解決すること非常に困難なると共に，之を理會することは，却て意外に容易なり．無理數の定義も亦此種の問題に屬せり．器械的に算式を把玩するを以て數學の能事畢れりとする者，固より斯の如き問題に關涉あるべからず．然れども一般の健全なる理解力及成熟せる判斷力を以て之に臨むときは，問題の要點を攫取すること決して難からず．

第八章に於て特に量の性質を詳說せるは，量と數との關係を明にして，以て常識と學問とを連結せんと欲せるなり．而して特に重をユークリツドの比例論に置けるは，啻に其重要なるが爲のみにあらずして，此クラシツクが本邦の普通敎育に於て今尙忠實に反復せられつゝあるにも由れり．卽ち一方に於てはユークリツドの理會を確實にすると共に，一方に於て之に資りて無理數の起源を明瞭ならしめんとせるなり．第十章の所說稍〻高きに過ぎたるが如しと雖，一たびこの最高の立脚點より瞰望するときは，第九章に說きたる無理數に關せる煩雜なる諸定理も又第十一章に論ぜる冪及對數の諸性質も，盡く之を一眸の中に收めて，歷々之を掌に指すが如くなるを得べし．斯の如き登臨を阻むこと東道の責を盡せる者と言ひ難からんか．

所載の事項其性質上一々出典を擧げ難し．就中其重要なるを選み附錄として卷末に添ふ，固より遺漏なきを保せず．

此書取材の範圍狹小にして，記する所多くは斷片なり．他日時間の餘裕を得て，初等數學の全般に涉り再び讀者に見ゆるの機あらんことを期す．

　明治三十七年六月東京に於て

著　者　識

新式算術講義

目次

第一章　自然數の起源 ············································································································································································································································································································

一-一〇

物を數ふること，物に順序を附くること，兩者の關係○順序數の原則四條，數の名，命數法の意義○カルヂナル數，物の數は數ふる順序に關係なし，自然數

第二章　四則算法 ············································································································································································································································································································

一〇-六二

加法，加法の應用上の意義，交換の法則，組み合はせの法則，減法の可能，減法の應用○乘法の意義，加法に對する分配の法則，組み合はせの法則，交換の法則，倍數，除法可能の條件○零の定義及其性質○多くの數の加法及乘法，ヂリクレーの證明○減法及除法に關する定理○冪及其算法○除法の擴張，數の展開，十進法○十進法に於ける四則の演算

第三章　負數，四則算法の再審 ············································································································································································································································································································

六三-九三

廣義に於ける整數の定義，其命名，アルキメデスの法則，數學的歸納法の原理○加法及其性質○正數及負數，減法の可能，絕對値○乘法及其性質，除法

第四章　整除に關する整數の性質 ············································································································································································································································································································

九三-一三八

整除，倍數，相合式及ガウスの記法，剩餘の擴張，相合式の性質○十進法に於ける特殊なる整數の倍數の鑑識○最小公倍數及最大公約數○二つの數の最小公倍數及最大公約數，ポアンソーの幾何學的說明○一次不定方程式，一般の解答の決定，オイラーの解法○素數及合成數，合成數の素數分解，エラトステネスの篩，素數の數に限りなし○素數分解の應用

第五章　分數 ············································································································································································································································································································

一三八-一七四

分數班の構成，分數班內の相等大小及加法減法，整數と分數班との內容の一致○通分，一般分數の相等大小及加法減法，旣約分數○分數班の總括，數の新系統，其特徵，分布の稠密なること及等分の可能○倍加及等分，最小公倍數及最大公約數○分數の比，比例式，分數の乘法，除法

第六章　分數に關する整數論的の硏究 ············································································································································································································································································································

一七四-二五三

最小公倍數及最大公約數○冪の定義の擴張，負の指數○素數分解の應用○分數を部分的分數に分解すること○與へられたる分母を有する旣約眞分數の數，ガウスの函數 $\varphi (n)$，其性質及算式○分數の展開，命數法，小數○循環小數の起源○フエルマーの定理の間接證明○小數の四則演算

第七章　四則算法の形式上不易 ············································································································································································································································································································

二五三-二八一

有理數，算法の形式上不易，問題の說明○順及逆の算法，其關係○減法の汎通及負數，正負整數の乘法○除法の汎通と分數，有理數四則，除法の例外○有理數の大小

第八章　量の連續性及無理數の起源 ············································································································································································································································································································

二八一-三二二

具體の量，抽象の量○量の原則，量の比較，加合及連續○「有理區域」，其性質，量の公約，公倍○量を計るとは何の謂ぞ○ユークリツドの法式，二つの場合○公約なき量の實例○ユークリツドの比の定義，比と有理數との相等及大小，二つの比の相等及大小○量と直線上の點との對照，稠密なる分布は連續に非ず，連續の定義○結論，數の原則

第九章　無理數 ············································································································································································································································································································

三二三-三七九

限りなく多くの數，上限及下限○基本定理○稠密なる分布，等分の可能，アルキメデスの法則は，凡て連續の法則に含蓄せらる○有理數の兩斷と無理數○無理數の展開，無限小數の意義○量を計ること及其數値の展開○展開せられたる數の大小の比較，展開の唯一なること○無理數の加法及其性質○加法の近似的演算○比例に關する定理，比例式解法，乘法及除法の意義○乘法及除法の性質○負數

第十章　極限及連續的算法 ············································································································································································································································································································

三八〇-四一四

集積點，極限，其定義及例○集積點に關する基本の定理○無限列數，極限存在の條件○極限と四則，無理數及其算法の第二の定義○連續的算法の定義，連續的算法の擴張○單調の變動，單調なる算法の轉倒

第十一章　冪及對數 ············································································································································································································································································································

四一四-四三六

冪根の存在，基數及び指數の變動に伴ふ冪根の變動，指數限りなく增大するとき冪根は限りなく $1$ に近迫す○冪の定義の擴張，有理の指數，無理の指數○對數，其性質○開平の演算

附錄

學用語對譯

目次終