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• Introductory Note.
• Book I.
• Chap. I.: Whether a Demonstrative Science Exists
• Chap. II.: What Knowing Is, What Demonstration Is, and of What It Consists
• Chap. III.: A Refutation of the Error Into Which Some Have Fallen Concerning Science and Demonstration
• Chap. IV.: The Meaning of ‘distributive,’ ‘essential,’ ‘universal’
• Chap. V.: From What Causes Mistakes Arise With Regard to the Discovery of the Universal. How They May Be Avoided
• Chap. VI.: Demonstration Is Founded On Necessary and Essential Principles
• Chap. VII.: The Premises and the Conclusion of a Demonstration Must Belong to the Same Genus
• Chap. VIII.: Demonstration Is Concerned Only With What Is Eternal
• Chap. IX.: Demonstration Is Founded Not On General, But On Special and Indemonstrable Principles; Nor Is It Easy to Know Whether One Really Possesses Knowledge Drawn From These Principles
• Chap. X.: The Definition and Division of Principles
• Chap. XI.: On Certain Principles Which Are Common to All Sciences
• Chap. XII.: On Questions, And, In Passing, On the Way In Which Sciences Are Extended
• Chap. XIII.: The Difference Between the Demonstration and Science of a Thing’s Nature and Those of Its Cause
• Chap. XIV.: The Figure Proper to Demonstrate Syllogism
• Chap. XV.: On Immediate Negative Propositions
• Chap. XVI.: On Ignorance Resulting From a Defective Arrangement of Terms In Mediate Propositions
• Chap. XVII.: On Ignorance Resulting From a Defective Arrangement of Terms In Immediate Propositions
• Chap. XVIII.: On Ignorance As Resulting From Defective Sense Perception
• Chap. XIX.: Whether the Principles of Demonstration Are Finite Or Infinite
• Chap. XX.: Middle Terms Are Not Infinite
• Chap. XXI.: In Negations Some Final and Ultimate Point Is Reached Where the Series Must Cease
• Chap. XXII.: In Affirmations Some Final and Ultimate Point Is Reached Where the Series Must Cease
• Chap. XXIII.: Certain Corollaries
• Chap. XXIV.: Whether Universal Or Particular Demonstration Is Superior
• Chap. XXV.: That Affirmative Is Superior to Negative Demonstration
• Chap. XXVI.: Direct Demonstration Is Superior to Reduction Per Impossible
• Chap. XXVII.: What Science Is More Certain and Prior, and What Less Certain and Inferior
• Chap. XXVIII.: What Constitutes One Or Many Sciences
• Chap. XXIX.: Concerning Many Demonstrations of the Same Thing
• Chap. XXX.: On Fortuitous Occurrences
• Chap. XXXI.: Sense Perception Cannot Give Demonstrative Science
• Chap. XXXII.: On the Difference of Principles Corresponding to the Difference of Syllogisms
• Chap. XXXIII.: The Distinction Between Science and Opinion
• Chap. XXXIV.: On Sagacity
• Book II.
• Chap. I.: On the Number and Arrangements of Questions
• Chap. II.: Every Question Is Concerned With the Discovery of a Middle Term
• Chap. III.: The Distinction Between Definition and Demonstration
• Chap. IV.: The Essence of a Thing Cannot Be Attained By Syllogism
• Chap. V.: Knowledge of the Essence Cannot Be Attained By Division
• Chap. VI.: The Essence Cannot Be Proved By the Definition of the Thing Itself Or By That of Its Opposite
• Chap. VII.: Whether the Essence Can In Any Way Be Proved
• Chap. VIII.: How the Essence Can Be Proved
• Chap. IX.: What Essences Can and What Cannot Be Proved
• Chap. X.: The Nature and Forms of Definition
• Chap. XI.: The Kinds of Causes Used In Demonstration
• Chap. XII.: On the Causes of Events Which Exist, Are In Process, Have Happened, Or Will Happen
• Chap. XIII.: On the Search For a Definition
• Chap. XIV.: On the Discovery of Questions For Demonstration
• Chap. XV.: How Far the Same Middle Term Is Employed For Demonstrating Different Questions
• Chap. XVI.: On Inferring the Cause From the Effect
• Chap. XVII.: Whether There Can Be Several Causes of the Same Thing
• Chap. XVIII.: Which Is the Prior Cause, That Which Is Nearer the Particular, Or the More Universal?
• Chap. XIX.: On the Attainment of Primary Principles
• Appendix. Prior Analytics. Book II.
• Chap. XXIII.: On Induction
• XXIV.: On Example

CHAP. VII.

The Premises and the Conclusion of a Demonstration must belong to the same genus

It is not possible to arrive at a demonstration by using for one’s proof a different genus from that of the subject in question; e.g. one cannot demonstrate a geometrical problem by means of arithmetic. There are three elements in demonstrations:—(1) the conclusion which is demonstrated, i.e., an essential attribute of some genus; (2) axioms or self-evident principles from which the proof proceeds; (3) the genus in question whose properties, i.e. essential attributes, are set forth by the demonstrations. Now the axioms which form the grounds of the demonstration may be identical for different genera; but in cases where the genera differ, as do arithmetic and geometry, it is not possible, e.g. to adapt an arithmetical demonstration to attributes of spatial magnitudes, unless such magnitudes happen to be numbers. That such transference is possible in certain connections I will explain later (cf. Chap. IX.).

Arithmetical demonstration is restricted to the genus with which it is properly concerned, and so with other sciences. Hence if a demonstration is to be transferred from one science to another the subjects must be the same either absolutely or in some respect. Otherwise such a transference is clearly impossible, for the extremes and the middle terms must necessarily belong to the same genus, for if not they would not be essentially but only accidentally predicable of the subject.

Hence one cannot shew by means of geometry that opposites are dealt with by a single science nor yet that two cubes when multiplied together produce another cube. Nor can one prove what belongs to one science by means of another except when one is subordinate to the other, as optics are to geometry and harmonics to arithmetic.

Neither is geometry concerned with the question of an attribute of line which does not inhere in it as such, and does not result from the special principles of geometry, as for instance the question whether the straight line is the most beautiful kind of line, or whether the straight line is the opposite of a circumference, for these qualities of beauty and opposition do not belong to line as a result of its particular genus, but because it has some qualities in common with other subjects.