Construction: Divide a given straight line internally in extreme and mean ratio.
Theorem: The areas of two triangles which have an angle of one equal to the angle of the other are to each other as the products of the sides including those angles.
Problem: Given a circle of unit diameter and the side of a regular inscribed polygon, find the side of a regular inscribed polygon of double the number of sides.
Theorem: The four bisectors of the four angles of a quadrilateral form a second quadrilateral, the opposite angles of which are supplementary.
Theorem: If on the diameter of a circle two points be taken equally distant from the centre, the sum of the squares of the distances of any point of the circumference from these two points is constant.
Problem: Find the locus of the point of intersection of the three altitudes of a triangle, given a fixed base, and constant angle at the vertex.
ENGLISH GRAMMAR.—Candidates must have a good knowledge of _English grammar_; they must be able to define the terms used therein; to define the parts of speech; to give inflections, including declension, conjugation and comparison; to give the corresponding masculine and feminine gender nouns; to give and apply the ordinary rules of syntax.
They must be able to parse correctly any ordinary sentence; giving the subject of each verb, the governing word of each objective case, the word for which each pronoun stands or to which it refers, the words between which each preposition shows the relation, precisely what each conjunction and each relative pronoun connects, what each adjective and adverb qualifies or limits, the construction of each infinitive, and generally to show a good knowledge of the function of each word in the sentence.
They must be able to correct in sentences or extracts any ordinary grammatical errors.
It is not required that any particular text-book shall be followed: but the definitions, parsing, and corrections must be in accordance with good usage and common sense.
The examinations may include questions similar to the following:
1. Define and give examples of: _a._ infinitive; _b._ indirect object; _c._ simple sentence.
2. Give the principal parts of: _a._ choose; _b._ crow; _c._ freeze; _d._ fly; _e._ burst.