題記：「知識百寶箱」系列是「寶仁工作室」為了實踐2021-2022年度工作願景，而特別設立。目的旨在加快「寶仁工作室」的轉型，全面成為以「知識型專欄」為基礎之「知識主導型」的網誌。期望以協助提升大眾的學術素養為信條，並配合STEM的發展。除了普及科學知識外，也負起潛移默化為大家的人生有所改變的重責大任。將以學術專題探討、學習筆記為內容主體，回饋社會，服務讀者。
内容介紹：本篇屬重啟篇章，會跟讀者分析港大入學試數學課程，本人將透過多份歷史資料，配合本人對高等數學課程的了解，客觀分析港大入學試數學課程的歷年沿革。以協助讀者對港大入學試數學課程的歷年沿革有基本了解，從而幫助有讀者可以選用合適的Past Paper去備試，爭取好成績。以及協助教育工作者在進行數學教育工作，有更多參考資源可供參考。
各位大家好，本篇為《2024年第26篇：香港高級程度會考數學科目分析（第一班數學篇）》的附錄，以中英雙語形式，整理並列出當年的港大入學試數學課程。建議讀者連同正篇一同閲讀，從而令大家更清晰了解歷年港大入學試數學課程之發展。
在本人撰寫正篇及附錄期間，曾研讀大量參考讀物，包括：
Ø 梁操雅和羅天佑合著的《香港考評文化的承與變：從強調篩選到反映能力》
Ø 歷年港大之Calendar和Students’ Handbook
Ø 歷年港大入學試之Handbook of
the Matriculation Examination
Ø 歷年港大之Regulations for
the senior and junior local examinations and for the matriculation examination
這些Calendar和Students’ Handbook，以及上述列出之參考讀物，可以在香港中央圖書館、港大參考圖館和中大圖書館中找到。這都幫助本人對港大入學試數學課程有如此透徹的了解，特此鳴謝。
附錄1：劍橋本地考試（Cambridge Local Examinations）及牛津本地考試（Oxford Local Examinations）1886-1912
1886 version
Junior
Mathematics (Section 7)
1.
Geometry
Euclid, Book i and ii
2.
Elementary Algebra
Definition and explanations of algebraical signs and terms; addition
subtraction multiplication and division of algebraical quantities; theory of
indices with integral exponentials; greatest common measure and least common
multiple; extraction of the square root; the solution of easy questions of the
first degree and questions producing such equations, and the solution of easy
quadratic equations involving one unknown quantity.
3.
Plane Geometry
Euclid, Book iii, iv and vi
4.
Algebra
Quadratic equations, the elementary rules of ratio and proportion,
arithmetical and geometrical progressions, permutations, combinations and the
binomial theorem with positive integral exponents
5.
Plane Trigonometry
The solution of triangles and the use of logarithms
6.
Elementary Mechanics
The composition and resolution of forces acting in one plane at a
point, parallel forces, the mechanical powers, the properties of the centre of
gravity, uniform and uniformly accelerated motion in a straight line.
Euclid and Algebra, if done well enough,
are sufficient for the mark of distinction.
Euclid’s axioms will be required, and no
proof of any proposition will be admitted which assumes the proof of anything
not proved in processing propositions in Euclid.
（中文參考譯文）
數學（第七部分）
1.
幾何
《Euclid》，Book i、ii
2.
初等代數
代數符號和術語的定義和解釋；代數量的加減乘除；整數指數理論；最大公因數和最小公倍數；提取平方根；解決簡單的一階問題和產生此類方程式的問題，以及解決涉及一個未知量的簡單二次方程式。
3.
平面幾何
《Euclid》，Book iii、iv及vi
4.
代數
二次方程式、比和比例的基本規則、等差和等比級數、排列、組合以及具有正整指數的二項式定理
5.
平面三角學
三角形的解法和對數的使用
6.
基本力學
作用在一個平面上一點的力的合成與解析、平行力、機械功率、重心的特性、勻速和勻加速的直線運動。
歐幾里德幾何和代數，如果做得夠好，就足以獲得優異（Distinction）。
將需要歐幾里德公理，並且不會接受任何假設證明在處理歐幾里德命題時未證明的任何命題的證明。
Senior
Section E
1.
Plane Geometry
Euclid, Book i, ii, iii, iv, vi and xi to Prop. 21 inclusive
2.
Algebra
The solution of simple and quadratic equations of and problems
producing such equations, the elementary rules of ratio proportion and
variation, arithmetical and geometrical progressions, permutations,
combinations, the binomial theorem, and the theory of logarithms.
See the remark of Euclid on p.3. Euclid and Algebra, if done well
enough, are sufficient for this mark of distinction.
3.
Plane Trigonometry
Inclusive of the Exponential Theorem, De Moivre’s Theorem, and the
expansion of sin θ and cos θ in powers of θ; the paper will also contain some easy questions on the more
advanced parts of algebra.
4.
Conic Sections
Treated both geometrically and by easy analytical geometry.
5.
Applied Mathematics
A.
Elementary Statics
The fundamental ideas of mass weight and density, the equilibrium of
forces acting in one plane, the properties of the centre of gravity, the laws
of friction, and the mechanical powers.
B.
Elementary Dynamics
The laws of motion and simple applications of them, uniform and
uniformly accelerated motion in a straight line, the laws of falling bodies,
projectiles, Atwood’s machine, and the principle of work with elementary
applications of it
C.
Elementary Parts of Astronomy
So far as they are necessary for the general explanation of the more
simple phenomena depending on the positions and motions of the bodies forming
the solar system.
（中文參考譯文）
第E部分
1.
平面幾何
《Euclid》，Book i、ii、iii、iv、vi和xi至第 21 條
2.
代數
簡單方程式和二次方程式的解法以及產生此類方程式的問題、比例和變分的基本規則、等差和等比級數、排列、組合、二項式定理和對數理論。
參見歐幾裡得在第3頁的評論。 歐幾里得幾何和代數，如果做得夠好，就足以達到這個優異（Distinction）水平。
3.
平面三角學
包括指數定理、棣美弗定理以及sinθ和cosθ的θ次方展開式；該試卷還將包含一些關於代數更高級部分的簡單問題。
4.
圓錐曲線
透過幾何和簡單的解析幾何進行處理。
5.
應用數學
A.
基本靜力學
質量重量和密度的基本概念、作用在一個平面上的力的平衡、重心的性質、摩擦定律和機械功率。
B.
基本動力學
運動定律及其簡單應用，勻速和勻加速直線運動，落體定律，拋射體，阿特伍德機器，以及工作原理及其基本應用
C.
天文學的基本部分
到目前為止，它們對於根據形成太陽系的天體的位置和運動來一般解釋更簡單的現象是必要的。
附錄2：港大入學試（Matriculation Examination）1913-1937
Version 1: 1913-1914
Mathematics (Group A: Obligatory
Subjects)
1.
Arithmetic
2.
Algebra, up to Quadratic
Equations
3.
Geometry, including the subject
matter of Euclid Books I, II
and III, with easy deductions.
Trigonometry (Group B: Optional Subjects)
Measurement of Angle in Degree, Minutes and
Seconds. Definition and Simple Relations of the Trigonometrical Ratios of a
Positive Angle not greater than two right angles. Simple Relations between the
Sides and Angles of a Triangle. Use of four-figure logarithmic Tables in Simple
Problems including the Solution of Triangles.
Mechanics (Group B: Optional Subjects)
Composition and Resolution of Forces and
Velocities acting in one Plane at a point. Parallel Forces, including simple
applications of Graphical Methods to such Forces. Moments. Centre of Gravity
and its determination in simple cases; applications to the Inclined Plane,
Lever, Common Balance, Wheel and Axle. Block and Tackle. Simple explanations
and illustrations of Friction. Rectilinear Motion under Uniform Acceleration.
Mass. Momentum. Force. Action and Reaction. A theoretical proof of the
Parallelogram of Forces will not be required.
（中文參考譯文）
數學（A組：必修科目）
1.
算術
2.
代數，直至二次方程
3.
幾何，包括《歐幾里得第一卷、第二卷和第三卷》的主題，並具有簡單的推論。
三角學（B 組：選修科目）
以度、分和秒為單位的角度測量。不大於兩個直角的正角的三角比的定義和簡單關係。三角形的邊和角之間的簡單關係。在簡單問題中使用四位對數表，包括解三角形。
力學（B 組：選修科目）
作用在一個平面上一點的力和速度的合成和分解。平行力，包括圖形方法對此類力的簡單應用。
力矩。 簡單情況下的重心及其確定；斜面、槓桿、普通平衡、輪軸的應用。滑輪組。摩擦力的簡單解釋和插圖。勻加速度下的直線運動。質量動量。力。行動和反應。不需要力的平行四邊形的理論證明。
Version 2: 1914-1915,
1917-1918, 1919-1920, 1920-1921, 1922, 1923, 1924
Mathematics (Group A: Obligatory
Subjects)
A.
Arithmetic.
(1914-1915)
General-Questions may be set in which contracted methods of
multiplication of division must be employed. Purely algebraic solutions will
not be accepted.
(since 1917-1918)
General-Questions may be set in which contracted methods of
multiplication of division must be employed. Algebraic solutions may be
employed, but all the steps of the reasoning must be clearly explained.
B.
Algebra.
(1914-1915)
This includes the course prescribed in Elementary Algebra and
Advanced Algebra for the Junior Examinations.
(since 1917-1918)
Elementary algebraical processes; symbolical representations; (1924
added) fractions; factors; square root, simple equations containing one or more
unknown quantities, quadratic equations containing one unknown quantity and
problems leading thereto. The solution of easy quadratic equations in two
unknown quantities; HCF and LCM; Surds; Ratio and Proportion; the elementary
properties and use of Logarithms; Simple Identities; Graphic solution of simple
simultaneous equations.
C.
Geometry.
As represented by the subject-matter of Schedule A(i), A(ii), B(i)
and B(ii).*
To pass in Mathematics Candidates must
reach a sufficient standard in all three divisions. (1923 added)
（中文參考譯文）
數學（A組：必修科目）
A.
算術
（1914-1915）
一般－可能會提出必須使用乘除法的簡化方法的問題。純代數解將不被接受。
（自1917-1918）
一般－可能會提出必須使用乘除法的簡化方法的問題。可以使用代數解，但必須清楚解釋推理的所有步驟。
B.
代數
（1914-1915）
這包括初級試中規定的初級代數和高級代數課程。
（自1917-1918）
基本代數過程；符號表示；（1924新增）分數；因數； 平方根、包含一個或多個未知量的簡單方程式、包含一個未知量的二次方程式、由此產生的問題。 兩個未知量的簡單二次方程式的解；最大公因數和最小公倍數；根式；比和比例；對數的基本特性和使用；簡單恒等式；簡單聯立方程式的圖解
C.
幾何。
如附表A(i)、A(ii)、B(i) 和 B(ii)的主題所示。*
要通過數學考試，考生必須在所有三個部分都達到足夠的標準。（1923新增）
Higher Mathematics (Group B: Optional
Subjects)
A.
Higher Geometry
(included the Elementary Geometry of the plane and sphere and the
Algebraical Geometry of the straight line, circle and conic section).*
B.
Advanced Algebra and Higher
Plane Trigonometry. (1914-1915)
Higher Algebra and Higher Plane Trigonometry (since 1917-1918)
Higher Algebra (since 1917-1918)
A more thorough knowledge of the course A2(b), together with the
following: Simultaneous Equations in three unknown quantities. Factorization;
Imaginary and Complex Quantities; The Progressions; Permutations and
Combinations, the Binomial Theorem for a positive integral exponent; Partial
Fractions; Elimination.
Higher Plane Trigonometry (since 1917-1918)
The periodicity of circular functions; inverse circular functions;
identities and transformations; expansions of functions of multiple angle;
relations between the circular functions and the circular measure of an angle;
the solution of triangles; properties of triangles and quadrilaterals; De
Moivre’s Theorem and its applications.
C.
Elementary Differential
Calculus.
Nature of the process of differentiations. Differentiation of 
, 
, 
and of easy rational functions. Rules for
differentiation of sums, products, quotients, functions of a function. Second
differentiation. Tangents and normal of simple curves. Velocities, including
rates of increase and diminution in general. Maxima and minima of simple
functions of one variable. Simple cases of approximation, e.g. calculation of
small corrections. Acceleration in simple cases.
D.
Applied Mathematics (Statics
and Dynamics).
Opportunity will be given for Candidates to show knowledge of
methods involving the use of the Differential and Integral Calculus.
One of the above divisions is sufficient
for a pass, no Candidate may take more than two divisions. (1914-1915 added)
（中文參考譯文）
高等數學（B組：選修科目）
A.
進階幾何
（包括平面和球面的初等幾何以及直線、圓和圓錐曲線的代數幾何）。*
B.
高階代數和進階平面三角學。（1914-1915）
進階代數和進階平面三角學（自1917-1918）
進階代數（自1917-1918）
更全面地了解課程 A2(b) [即數學Mathematics代數部分]，以及以下內容：三個未知量的聯立方程式。因式分解；虛數和複數；數列；排列與組合、正整指數的二項式定理；部分分式；消去法。
進階平面三角學（自1917-1918）
三角函數的週期性；反三角函數；恒等式與轉換；多倍函數的展式；三角函數和角度的角量度之間的關係；三角形的解；三角形和四邊形的性質；棣美弗定理及其應用。
C.
初等微積分
微分過程的本質。、、 和簡單有理函數的微分。和、積、商、函數的函數的微分法則。二階微分。簡單曲線的切線和法線。速度，包括一般的增加率和減少率。
單變數簡單函數的最大值和最小值。簡單的近似個案，例如
微增量的計算。簡單情況下的加速度。
D.
應用數學（靜力學和動力學）
上述一個部分足以通過，任何候選人不得超過兩個部分。（1914-1915 年新增）
Trigonometry (Group B: Optional Subjects)
(1914-1915 to 1923)
Measurement of Angle in Degree, Minutes and
Seconds. Definition and Simple Relations of the Trigonometrical Ratios of a
Positive Angle not greater than two right angles. Simple Relations between the
Sides and Angles of a Triangle. Use of four-figure logarithmic Tables in Simple
Problems including the Solution of Triangles.
(1924)
Measurement of Angle in Degree, Minutes, Seconds
and Radians. Definition and Simple Relations of the Trigonometrical Ratios of a
Positive Angle not greater than two right angles. Simple Relations between the
Sides and Angles of a Triangle. Use of four-figure logarithmic Tables in Simple
Problems including the Solution of Triangles.
（中文參考譯文）
三角學（B 組：選修科目）
（1914-1915至1923）
以度、分和秒為單位的角度測量。不大於兩個直角的正角的三角比的定義和簡單關係。三角形的邊和角之間的簡單關係。在簡單問題中使用四位對數表，包括解三角形。
（1924）
以度、分、秒和弧度為單位的角度測量。不大於兩個直角的正角的三角比的定義和簡單關係。三角形的邊和角之間的簡單關係。在簡單問題中使用四位對數表，包括解三角形。
Mechanics (Group B: Optional Subjects)
(1914-1915, 1917-1918)
Composition and Resolution of Forces and
Velocities acting in one Plane at a point. Parallel Forces, including simple
applications of Graphical Methods to such Forces. Moments. Centre of Gravity
and its determination in simple cases; applications to the Inclined Plane,
Lever, Common Balance, Wheel and Axle. Block and Tackle. Simple explanations
and illustrations of Friction. Rectilinear Motion under Uniform Acceleration.
Mass. Momentum. Force. Action and Reaction. A theoretical proof of the
Parallelogram of Forces will not be required.
(since 1919-1920)
Elementary notion of displacement, Velocity
and Acceleration. Motion of a body with constant Acceleration. Resolution and
Composition of Velocities, Accelerations, etc. Elementary notions of Mass and Momentum.
Elementary notions of Force as measured by rate of change of Momentum. Newton’s
Laws of Motion. Kinetic Energy, and Work. Units of Force, and Measurement.
Balancing of Forces. Torques or Moments. Conditions for the equilibrium of
Three Parallel Forces. Resolution and Composition of Parallel Forces in one
Plane. Centre of Parallel Forces. Centre of Gravity. Stable, Unstable, and
Neutral Equilibrium. Conditions for the equilibrium of Three Forces not
parallel. Triangle and Parallelogram of Forces. Moments. Simple illustrations
of Conditions of Equilibrium and of the Principle of Work, as in levers,
pulleys, the inclined plane, etc. Pressure in Liquids; variations with depth.
Transmission of Liquid Pressure; Hydraulic Press. Pressure on immersed and
floating bodies. Density; methods of determining Relative Densities. Relation
between volume and pressure in Gases. Atmospheric Pressure.
（中文參考譯文）
力學（B 組：選修科目）
（1914-1915、1917-1918）
作用在一個平面上一點的力和速度的合成和分解。平行力，包括圖形方法對此類力的簡單應用。
力矩。 簡單情況下的重心及其確定；斜面、槓桿、普通平衡、輪軸的應用。滑輪組。摩擦力的簡單解釋和插圖。勻加速度下的直線運動。質量動量。力。行動和反應。不需要力的平行四邊形的理論證明。
（自1919-1920）
* General Instructions for Geometry
The papers in Geometry will contain
questions on Practical and Theoretical Geometry. Every Candidate will be
expected to answer questions in both branches of the subject.
The questions on Practical Geometry will be
set on the constructions contained in the annexed Schedule A, together with
easy extensions of them. In cases where the validity of a construction is not
obvious, the reasoning by which it is justified may be required. Every
Candidates must provide himself with a ruler graduated in inches and tenths of
an inch, and in centimetres and milimetres, a set square, a protractor,
compasses, and a fairly hard pencil. All figures must be drawn accurately and
distinctly. Questions may be set in which the use of the set square or of the
protractor is forbidden.
The questions on Theoretical Geometry will
consist of theorems contained in the annexed Schedule B, together with
questions upon these theorems, easy deductions from them, and arithmetical
illustrations, Any proof of a proposition will be accepted which appears to the
examiners to form part of a systematic treatment of the subject; the order in
which the theorems are stated in Schedule B is not imposed as a sequence of
their treatment. In the proof of theorems and deductions from them, the use of
hypothetical constructions will be permitted.
Schedule A – Practical
A(i)
1.
Bisection of angles and of
straight lines.
2.
Construction of perpendiculars
to straight lines.
3.
Construction of parallels to a
given straight line.
4.
Simple cases of the
construction of triangles from given data.
5.
Division of straight lines into
a given number of equal parts or into parts in any given proportions.
6.
Construction of tangents to a
circle and of common tangents to two circles.
7.
Construction of circumscribed,
inscribed and escribed circles of a triangle.
A(ii)
1.
Simple cases of the
construction of circles from sufficient data.
2.
Construction of regular figures
of 3, 4, 6, or 8 sides in or about a given circle.
3.
Construction of a triangle
equal in area to given polygon.
4.
Construction of a square equal
in area to a given polygon. (1922 cancelled)
5.
Construction of rectilineal
figures from given data.
6.
Construction of solve
graphically a quadratic equation.
7.
Construction to divide a given
line in medial section.
Schedule B – Theoretical
B(i)
Angles at a point*
1.
If a straight line stands on
another straight line, the sums of the two angles no formed is equal to two
right angles; and the converse.
2.
If two straight lines
intersect, the vertically opposite angles are equal.
Parallel Straight Lines
1.
When a straight line cuts two
other straight lines,
(i) a pairs alternate angles are equal,
or (ii) a pair of corresponding angles are equal,
or (iii) a pair of interior angles on the same side of the cutting
line are together equal to two right angles,
then the two straight lines are parallel; and the converse.
2.
Straight lines which are
parallel to the same straight line are parallel to one another.
Triangles and Rectilinear Figures
1.
The sum of the angles of a
triangle is equal to two right angles.
2.
If sides of a convex polygon
are produced in order, the sum of the angles so formed is equal to four right
angles.
3.
If two triangles have two sides
of the one equal to two sides of the other, each to each, and also the angles
contained by those sides equal, the triangles are congruent.
4.
If two triangles have two angles
of the one equal to two angles of the other, each to each, and also one side of
the one equal to the corresponding side of the other, the triangles are
congruent.
5.
If two sides of a triangle are
equal, the angles opposite to these sides are equal, and the converse,
6.
If two triangles have the three
sides of the one equal to the three sides of the other, each to each, the
triangles are congruent.
7.
If two right-angled triangles
have their hypotenuses equal, and one sides of the one equal to one side of the
other, the triangles are congruent.
8.
If two sides of a triangle are
unequal, the greater side has the greater angle opposite to it; and the
converse.
9.
Of all the straight lines that
can be drawn to a given straight line from a given point outside it, the
perpendicular is the shortest.
10.
The opposite sides and angles
of a parallelogram are equal, each diagonal bisects the parallelogram, and the
diagonals bisect one another.
11.
If there are three or more
parallel straight lines, and the intercepts made by them on any straight line
that cuts them are equal, then the corresponding intercepts on any other
straight line that cuts them are also equal.
Areas.
1.
Parallelograms on the same or
equal bases and of the same altitude are equal in area.
2.
Triangles on the same or equal
bases and of the same altitude are equal in area.
3.
Equal triangles on the same or
equal bases are of the same altitude.
4.
The square on the hypotenuse of
a right-angled triangle is equal to the sum of the squares on the other two
sides; and the converse.
The Circle.
1.
A straight line, drawn from the
centre of a circle to bisect a chord which is not a diameter, is at right
angles to the chord; conversely, the perpendicular to a chord from the centre
bisects the chord.
2.
There is one circle, and one
only, which passes through three given points not in a straight line.
3.
Equal chords of a circle are
equidistant from the centre; and the converse.
4.
The tangent at any point of a
circle and the radius through the point are perpendicular to one another.
5.
If two circles touch, the point
of contact lies on the straight line through the centres.
6.
The angle which an are of a
circle subtends at the centre is double that which is subtends at any point on
the remaining part of the circumference.
7.
Angles in the same segment of a
circle are equal; and, if the line joining two points subtends equal angles at
two other points on the same side of it, the four points lie on a circle.
8.
The angle in a semicircle is a
right angle; the angle in a segment greater than a semicircle is less than a
right angle; and the angle is a segment less than a semicircle is greater than
a right angle.
9.
The opposite angles of any quadrilateral
inscribed in a circle are supplementary; and the converse.
B(ii)
1.
Illustrations and explanations
of the geometrical theorems corresponding to the following algebraical
identities:
2.
The square on a side of a
triangle is greater or less than the sum of the squares on the other two sides,
according as the angle contained by those sides is obtuse or acute. The
difference is twice the rectangle contained by one of the two sides and the
projection on it of the other.
3.
In equal circles (or, in the
same circle) (i) if two arcs subtend equal angles at the centres, they are
equal; (ii) conversely, if two arcs are equal, they subtend equal angles at the
centres.*
4.
In equal circles (or, in the
same circle) (i) if two chords are equal, they cut off equal arcs; (ii)
conversely, if two arcs are equal, the chords of the arcs are equal.*
5.
If a straight line touches a
circle, and from the point of contact a chord is drawn, the angles which this
chord makes with the tangent are equal to the angles in the alternate segments.
6.
If two chords of a circle
intersect either inside or outside the circle, the rectangle contained by the
parts of the one is equal to the rectangle contained by the parts of the other.
Loci.
1.
The locus of a point which is
equidistant from two fixed points is the perpendicular bisector of the straight
line joining the two fixed points.
2.
The locus of a point which is
equidistant from two intersecting straight lines consists of the pair of
straight lines which bisect the angles between the two given lines.
3.
The loci of points which move
so that the sum or difference of their distances from two fixed points or from
two given lines are constant.
4.
The construction of triangles
from given data by the methods of the intersection of loci.
* Proofs of these propositions will not be
required.
（中文參考譯文）
*幾何一般說明
幾何試卷將包含實用幾何和理論幾何的問題。
每個考生都需要回答該主題的兩個分支的問題。
實用幾何問題將根據附件附表A中包含的結構以及它們的簡單擴充進行設定。如果解釋的有效性不明顯，則可能需要證明其合理性的推理。每個考生必須為自己準備一把以英吋和十分之一英吋為單位、以公分和毫米為單位的尺、一個三角尺、一個量角器、圓規和一支硬身鉛筆。所有圖形必須準確清晰地繪製。可能會提出禁止使用三角板或量角器的問題。
理論幾何問題將包括附表B中包含的定理，以及關於這些定理的問題、從中進行的簡單推論以及算術插圖。任何在考官看來構成命題一部分的命題證明都將被接受。受試者的系統治療；附表B中陳述定理的順序並未強加為處理它們的順序。在證明定理和推論時，允許使用假設結構。
附表 A –
實用
A(i)
1.
角和直線的平分。
2.
垂直於直線的繪製。
3.
構造與給定直線的平行線。
4.
繪製給定資料構造三角形的簡單情況。
5.
將直線劃分為給定數量的相等部分或按任意給定比例劃分。
6.
繪製圓的切線和兩個圓的公切線。
7.
三角形的外接圓、內切圓和外切圓的繪製。
A(ii)
1.
根據足夠的數據建立圓形的簡單個案。
2.
在給定圓內或圍繞給定圓繪製3、4、6 或 8 條邊的規則圖形。
3.
繪製面積等於給定多邊形的三角形。
4.
繪製面積與給定多邊形相等的正方形。（1922取消）
5.
根據給定資料建立直線圖形。
6.
繪製以圖形方式求解二次方程式。
7.
在內側部分劃分給定線的繪製。
附表 B –
理論
B(i)
直線上的角*
1.
若一條直線與另一條直線重疊，則兩條直線所成的角和等於兩個直角；
反之亦然。[直線上的鄰角]
2.
如果兩條直線相交，則垂直之對角相等。[對頂角]
平行直線
1.
當一條直線與另外兩條直線相交時，
(i) 一對内錯角相等，
或 (ii) 一對同位角相等，
(iii) 截線同一側的一對內角合在一起等於兩個直角，
那麼兩條直線平行； 反之亦然。[内錯角相等、同位角相等、同旁内角互補]
2.
與同一條直線平行的直線彼此平行。
三角形和直線圖形
1.
三角形的內角和等於兩個直角。[三角形內角和]
2.
如果依序產生凸多邊形的邊，則所形成的角和等於四個直角。[多邊形外角和]
3.
如果兩個三角形的一邊與另一個三角形的兩邊相等，而這些邊所含的角也相等，則這兩個三角形全等。[SAS]
4.
如果一個三角形的兩個角等於另一個三角和的兩個角，且一個三角形的一邊等於另一個三角形的對應邊，則這兩個三角形全等。[ASA、AAS]
5.
如果三角形的兩邊相等，則與這些邊相對的角也相等，反之亦然，
6.
如果一個三角形的三邊等於另一個三角形的三邊，則這兩個三角形全等。[SSS]
7.
如果兩個直角三角形的斜邊相等，且直角三角形的一側等於另一個直角三角形的一側，則這兩個三角形全等。[RHS]
8.
若三角形的兩邊不相等，則較大邊的所對的角較大；反之亦然。[大邊對大角、大角對大邊]
9.
在從給定直線外部的給定點可以繪製到給定直線的所有直線中，垂線是最短的。
10.
平行四邊形的對邊和角相等，每條對角線平分平行四邊形，對角線互相平分。[平行四邊形對角、平行四邊形對邊、平行四邊形對角線]
11.
如果有3條或3條以上平行直線，且它們在與它們相交的任何一條直線上的截距相等，則在與它們相交的任何其他直線上相應的截距也相等。[截線定理]
面積
1.
相同或相等底且相同高度的平行四邊形的面積相等。
2.
相同或相等底、相同高的三角形面積相等。
3.
相同或相等的底上的相等三角形具有相同的高度。
4.
直角三角形斜邊的平方等於另外兩邊的平方和；反之亦然。[畢氏定理]
圓
1.
由圓心平分非直徑弦的直線與弦成直角；相反，從中心到弦的垂線平分該弦。[圓心至弦的垂線平分弦、圓心至弦中點的連線垂直弦]
2.
有一個圓，而且只有一個圓，它穿過三個給定的點，但不在一條直線上。
3.
圓的等弦距圓心等距；反之亦然。[等弦對等弦心距、等弦心距對等弦]
4.
圓上任一點的切線和經過該點的半徑互相垂直。[切線⊥半徑]
5.
如果兩個圓相接觸，則接觸點位於通過圓心的直線上。
6.
圓的圓心所對的角度是圓周其餘部分任意點所對的角度的兩倍。[圓心角兩倍於圓周角]
7.
圓內同一弓形的角相等；並且，如果連接兩點的線與同一側的另外兩點所成的角度相等，則這四個點位於一個圓上。[同弓形內的圓周角、同弓形內的圓周角的逆定理]
8.
半圓內的角是直角；
大於半圓的線段內的角小於直角； 角度是小於半圓大於直角的線段。[半圓上的圓周角]
9.
圓內切四邊形的對角互補；反之亦然。[圓內接四邊形對角、對角互補]
B(ii)
1.
對應於以下代數恆等式的幾何定理的圖示與解釋：
[乘法分配律、完全平方、平方差]
2.
三角形的一邊的平方大於或小於另外兩邊的平方和，取決於這些邊所含的角度是鈍角還是銳角。
其差值是兩邊之一和另一邊在其上的投影所包含的矩形的兩倍。
3.
在等圓中（或在同一個圓中） (i) 如果兩個圓弧的圓心所對的角度相等，則它們相等； (ii) 相反，如果兩條弧相等，則它們在中心處所對的角度相等。*[等角對等弧、等弧對等角]
4.
在等圓中（或在同一個圓中） (i) 如果兩個弦相等，則它們截出的弧相等； (ii) 相反，若兩條弧相等，則這兩條弧的弦也相等。*[等弦對等弧、等弧對等弦]
5.
如果一條直線與圓相切，並從接觸點畫一條弦，則該弦與切線所成的角度等於交替線段中的角度。[交錯弓形上的圓周角]
6.
如果圓的兩條弦在圓的內部或外部相交，則其中一個的部分所包含的矩形等於另一個的部分所包含的矩形。
軌跡
1.
與兩個固定點等距的點的軌跡是連接兩個固定點的直線的垂直平分線。
2.
與兩條相交直線等距的點的軌跡由平分兩條給定線之間的角度的一對直線組成。
3.
移動的點的軌跡，使得它們距離兩個固定點或兩條給定線的距離總和或差恆定。
4.
透過軌跡交集的方法根據給定資料建立三角形。*
*不需要這些命題的證明。
Version 3: 1925, 1926,
1927, 1928
Mathematics (Group A: Obligatory Subject)
1.
Arithmetic.
General-Questions may be set in which contracted methods of
multiplication of division must be employed. Algebraic solutions may be
employed, but all the steps of the reasoning must be clearly explained.
2.
Algebra.
Elementary algebraical processes; symbolical representation; fractions;
factors; square root, simple equations containing one or more unknown
quantities, quadratic equations containing one unknown quantity and problems
leading thereto. The solution of easy quadratic equations in two unknown
quantities; HCF and LCM; Surds; Ratio and Proportion; the elementary properties
and use of Logarithms; Simple Identities; Graphic solution of simple
simultaneous equations; the Progressions.
3.
Geometry.
(1925 to 1927)
The papers in Geometry will contain questions on both Theoretical
and Practical Geometry.
No text-book is
specified, and none need be quoted in answering questions, but the requirements
are covered by Hall and Stevens “School Geometry” Part I, II and III.
(1928)
The general geometry of the triangle and its associated circles.
Areas of triangles, parallelograms, trapeziums, segments and sectors of
circles. Simple loci. Properties of similar triangles and other figures.
Proportion. Easy deductions.
All proofs of geometrical propositions must be geometrical.
The papers set will contain questions on both Theoretical and
Practical Geometry.
No text-book is
specified, and more need be quoted in the answers but the requirements are
covered by Hall and Stevens’ “School Geometry” up to page 310: (i.e. Part I,
II, III, IV and portion of V).
To pass in Mathematics Candidates must
reach a sufficient standard in all three divisions.
（中文參考譯文）
數學（A組：必修科目）
1.
算術
一般－可能會提出必須使用乘除法的簡化方法的問題。 可以使用代數解，但必須清楚解釋推理的所有步驟。
2.
代數
基本代數過程；符號表示；分數；因數； 平方根、包含一個或多個未知量的簡單方程式、包含一個未知量的二次方程式、由此產生的問題。兩個未知量的簡單二次方程式的解；最大公因數和最小公倍數；根式；比和比例；對數的基本特性和使用；簡單恒等式；簡單聯立方程式的圖解；數列。
3.
幾何。
（1925年至1927年）
沒有指定教科書，也不需要在回答問題時引用任何教科書，但Hall和Stevens之《School Geometry Part I,
II, III》涵蓋了這些要求。
（1928）
三角形及其相關圓的一般幾何形狀。三角形、平行四邊形、梯形、圓的線段和扇形的面積。簡單軌跡。相似三角形和其他圖形的性質。比例。簡易演繹法。
所有幾何命題的證明都必須是幾何的。
試卷將包含理論幾何和實用幾何的問題。
沒有指定教科書，答案中需要引用更多內容，但Hall和Stevens之《School Geometry》直至第310頁，涵蓋了這些要求：（即第一部分、第二部分、第三部分、第四部分和部分之第五部分）。
要通過數學考試，考生必須在所有三個部分都達到足夠的標準。
Higher Mathematics (Group B: Optional
Subject)
1.
Higher Algebra.
A more thorough knowledge of the course 2(b), together with the
following: Simultaneous Equation in three unknown quantities. Theory of
Quadratic Equations; Imaginary and Complex Quantities. Permutations and
Combinations; the Binomial Theorem; Partial Fractions; Exponential and
Logarithmic Series.
2.
Higher Trigonometry.
The periodicity of circular functions; inverse circular functions;
identities and transformations; expansions of functions of multiple angle;
relations between the circular functions and the circular measure of an angle;
the solution of triangles; properties of triangles and quadrilaterals; De
Moivre’s Theorem and its applications; Elimination.
3.
Elementary Calculus.
Nature of the process of differentiation. Differentiation of 
, , , and 
and of easy rational functions of a function.
Second differentiation. Velocities and rates of increase and diminution in
general. Maxims and minima of simple functions of one variable. Simple cases of
approximation, e.g. calculation of small correction. Integration as converse of
differentiation. Definite and indefinite integrals of , , , 
. Easy applications of the Calculus.
Tangents and normal to curves, simple areas and volumes.
4.
Higher Plane and Solid
Geometry.
Including the more advanced geometry of the triangle and its
associated circles, and the elementary geometry of the plane and sphere, and
the simpler regular solids.
5.
Coordinate Geometry.
Straight line and circle.
6.
Conics.
Coordinate geometry of the parabola, ellipse, and hyperbola.
7.
Applied Mathematics. (i.e.
Dynamics, including Kinematics, Statics and Kinetics).
Candidates will be expected to solve problems involving use of the
Diff. and Int. Calculus.
One of the above divisions is sufficient
for a pass, no Candidate may take more than two divisions.
（中文參考譯文）
高等數學（B組：選修科目）
1.
進階代數
對課程2(b)[即Mathematics一科Algebra部分]以及以下內容有更全面的了解：三個未知量的聯立方程式。
二次方程式理論；虛數和複數。
排列和組合；二項式定理；部分分式；指數和對數級數。
2.
進階三角學
三角函數的週期性；反三角函數；恒等式與轉換；多倍函數的展式；三角函數和角度的角量度之間的關係；三角形的解；三角形和四邊形的性質；棣美弗定理及其應用；消去法。
微分過程的本質。、、、和簡單有理函數的微分。二階微分。速度及一般的增加率和減少率。單變數簡單函數的最大值和最小值。簡單的近似個案，例如微增量的計算。積是微分的逆運算。、、、 的定積分和不定積分。微積分的簡單應用。曲線的切線和法線、簡單面積和體積。
4.
進階平面和立體幾何
包括更高級的三角形及其相關圓的幾何形狀，以及平面和球體的基本幾何形狀，以及更簡單的正多面體。
5.
坐標幾何
直線和圓。
6.
圓錐曲線
拋物線、橢圓和雙曲線的座標幾何。
7.
應用數學。（即動力學，包括運動學、靜力學和動力學）
考生需要解決涉及使用微分和積分的問題。
上述一個部分足以通過，任何考生不得超過兩個部分。
Trigonometry (Group B: Optional Subjects)
Measurement of Angle in Degree, Minutes,
Seconds and Radians. Definition and Simple Relations of the Trigonometrical
Ratios of a Positive Angle not greater than two right angles. Simple Relations
between the Sides and Angles of a Triangle. Use of four-figure logarithmic
Tables in Simple Problems including the Solution of Triangles. Easy
trigonometric equations.
（中文參考譯文）
三角學（B 組：選修科目）
以度、分、秒和弧度為單位的角度測量。不大於兩個直角的正角的三角比的定義和簡單關係。三角形的邊和角之間的簡單關係。在簡單問題中使用四位對數表，包括解三角形。
Mechanics (Group B: Optional Subjects)
Elementary notion of displacement, Velocity
and Acceleration. Motion of a body with constant Acceleration. Resolution and
Composition of Velocities, Accelerations, etc. Elementary notions of Mass and Momentum.
Elementary notions of Force as measured by rate of change of Momentum. Newton’s
Laws of Motion. Kinetic Energy, and Work. Units of Force, and Measurement.
Balancing of Forces. Torques or Moments. Conditions for the equilibrium of
Three Parallel Forces. Resolution and Composition of Parallel Forces in one
Plane. Centre of Parallel Forces. Centre of Gravity. Stable, Unstable, and
Neutral Equilibrium. Conditions for the equilibrium of Three Forces not
parallel. Triangle and Parallelogram of Forces. Moments. Simple illustrations
of Conditions of Equilibrium and of the Principle of Work, as in levers,
pulleys, the inclined plane, etc. Pressure in Liquids; variations with depth.
Transmission of Liquid Pressure; Hydraulic Press. Pressure on immersed and
floating bodies. Density; methods of determining Relative Densities. Relation
between volume and pressure in Gases. Atmospheric Pressure.
（中文參考譯文）
力學（B 組：選修科目）
位移、速度和加速度的基本概念。物體以恆定加速度運動。速度、加速度等的分析與合成。質量與動量的基本概念。透過動量變化率測量的力的基本概念。牛頓運動定律。動能和作功。力的單位和測量。力的平衡。扭力或力矩。三個平行力平衡的條件。一個平面內平行力的分解與合成。平行力中心。重心。穩定、不穩定和中性平衡。三力平衡的條件不平行。三角形和平行四邊形的力。力矩。平衡條件和工作原理的簡單說明，如槓桿、滑輪、斜面等。隨深度的變化。液體壓力傳輸；液壓機。浸入水中和漂浮物體上的壓力。密度；測定相對密度的方法。氣體體積和壓力之間的關係。大氣壓力。
Version 4: 1929, 1930,
1931, 1932, 1933 (content don’t have any change)
Arrangement for Mathematical Subjects (since
1931):
Note 1
Obligatory Mathematics consists of either
(i) Arithmetic, Algebra and Geometry, or
(ii) Trigonometry, Algebra and Geometry
To pass in Obligatory Mathematics, a
Candidate must reach a sufficient standard in all three sections.
(a) Students who intend to take classes in
the University in Mathematics and/or Physics as part of a course qualifying for
a degree in Engineering or Arts must satisfy the Examiners in Trigonometry
either as part of the Obligatory Mathematics group or as a subject under
Optional Mathematics.
(b) Any Candidate who takes all four
subjects, Arithmetic, Algebra, Geometry and Trigonometry, shall, subject to
condition (a) above, be deemed to have passed on Obligatory Mathematics if he
satisfies the Examiners in either group (i) or group (ii).
(c) Any Candidate who passes in all four
subjects mentioned in (b) shall be deemed to have passed in Obligatory
Mathematics, and also in Trigonometry under Optional Mathematics. (Note 4
below).
Note 4
Optional Mathematics consists of
(i) Trigonometry, if not already included
in Obligatory Mathematics. (See Note 1 (c) above).
See under Higher Mathematics.
(ii) Mechanics. (1933 not include this
part)
(iii) Co-ordinate Geometry.
(iv) Higher Algebra.
(v) Higher Trigonometry.
(vi) Higher Plane and Solid Geometry.
(vii) Elementary Calculus.
(viii) Conics.
(ix) Applied Mathematics.
A Candidate may offer not more than four
sections from this group, and, if he satisfies the Examiners, receive credit
for them as endorsements on his certificate, but not more than two will be
reckoned as contributing towards his pass in matriculation.
（中文參考譯文）
數學科目安排（1931年起）：
註1
必修數學包括
(i) 算術、代數和幾何，或
(ii) 三角學、代數和幾何
要通過必修數學，考生必須在所有三個部分達到足夠的標準。
(a) 打算在大學修讀數學和/或物理課程作為有資格獲得工程或藝術學位的課程的一部分的學生必須讓三角學作為必修數學組的一部分或作為選修數學。
(b) 任何修讀所有四門科目（算術、代數、幾何和三角）的考生，在符合上述條件(a) 的情況下，如果其滿足(i) 組或(i) 組中考官的要求，則應視為已通過必修數學。
(c) 任何通過 (b) 中提到的所有四門科目的考生應被視為通過必修數學以及選修數學下的三角學。（下面的註 4）。
註4
選修數學包括
(i) 三角學（如果尚未包含在必修數學中）。（參閱上註 1 (c)）。
參見高等數學。
(ii) 力學。（1933年不包括這部分）
(iii) 坐標幾何。
(iv) 進階代數。
(v) 進階三角學。
(vi) 進階平面和立體幾何。
(vii) 初等微積分。
(viii) 圓錐曲線。
(ix) 應用數學。
考生可以從該組中提供不超過四個部分，並且，如果他滿足考官的要求，則可以在其證書上獲得這些部分的學分，但不得超過兩個部分將被視為有助於其通過入學考試。
Mathematics (Group A: Obligatory Subject)
Part 1: Arithmetic/ Trigonometry
1.
Arithmetic.
General-Questions may be set in which contracted methods of
multiplication of division must be employed. Algebraic solutions may be
employed, but all the steps of the reasoning must be clearly explained.
or
2.
Trigonometry.
Measurement of Angle in degree, minutes, seconds and radians. Definition
and Simple Relations of the Trigonometrical Ratios of a Positive Angle not
greater than two right angles. Simple Relations between the Sides and Angles of
a Triangle. Use of four-figure logarithmic Tables in Simple Problems including
the Solution of Triangles. Easy trigonometric equations.
Part 2: Algebra.
Ø Elementary algebraical processes; symbolical representation; fractions;
factors; square root, simple equations containing one or more unknown
quantities, quadratic equations containing one unknown quantity and problems
leading thereto. The solution of easy quadratic equations in two unknown
quantities; HCF and LCM; Surds; Ratio and Proportion; the elementary properties
and use of Logarithms; Simple Identities; Graphic solution of simple
simultaneous equations; the Progressions.
Part 3: Geometry.
1.
The general geometry of the
triangle and its associated circles. Areas of triangles, parallelograms,
trapeziums, segments and sectors of circles. Simple loci. Properties of similar
triangles and other figures. Proportion. Easy deductions.
2.
All proofs of geometrical
propositions must be geometrical.
3.
The papers set will contain
questions on both Theoretical and Practical Geometry.
4.
No text-book is specified, and more need be quoted in the
answers but the requirements are covered by Hall and Stevens’ “School Geometry”
up to page 310: (i.e. Part I, II, III, IV and portion of V).
Under (a) candidates may offer either (1)
Arithmetic or (2) Trigonometry.
To pass in Mathematics Candidates must
reach a sufficient standard in all three divisions.
（中文參考譯文）
數學（A組：必修科目）
第一部分：算術/三角學
1.
算術
一般－可能會提出必須使用乘除法的簡化方法的問題。 可以使用代數解，但必須清楚解釋推理的所有步驟。
或者
2.
三角學。
以度、分、秒和弧度為單位的角度測量。不大於兩個直角的正角的三角比的定義和簡單關係。三角形的邊和角之間的簡單關係。在簡單問題中使用四位對數表，包括解三角形。
第二部分：代數
Ø 基本代數過程；符號表示；分數；因數； 平方根、包含一個或多個未知量的簡單方程式、包含一個未知量的二次方程式、由此產生的問題。兩個未知量的簡單二次方程式的解；最大公因數和最小公倍數；根式；比和比例；對數的基本特性和使用；簡單恒等式；簡單聯立方程式的圖解；數列。
第三部分：幾何。
1.
三角形及其相關圓的一般幾何形狀。三角形、平行四邊形、梯形、圓的線段和扇形的面積。簡單軌跡。相似三角形和其他圖形的性質。比例。簡易演繹法。
2.
所有幾何命題的證明都必須是幾何的。
3.
試卷將包含理論幾何和實用幾何的問題。
4.
沒有指定教科書，答案中需要引用更多內容，但Hall和Stevens之《School Geometry》直至第310頁，涵蓋了這些要求：（即第一部分、第二部分、第三部分、第四部分和部分之第五部分）。
在 (a) 項[即第一部分]下，考生可以選擇 (1) 算術或 (2) 三角學。
要通過數學考試，考生必須在所有三個部分都達到足夠的標準。
Higher Mathematics (Group B: Optional
Subject)
1.
Higher Algebra.
A more thorough knowledge of the course 2(b), together with the
following: Simultaneous Equation in three unknown quantities. Theory of
Quadratic Equations; Imaginary and Complex Quantities. Permutations and
Combinations; the Binomial Theorem; Partial Fractions; Exponential and
Logarithmic Series.
2.
Higher Trigonometry.
The periodicity of circular functions; inverse circular functions;
identities and transformations; expansions of functions of multiple angle;
relations between the circular functions and the circular measure of an angle;
the solution of triangles; properties of triangles and quadrilaterals; De
Moivre’s Theorem and its applications; Elimination.
3.
Elementary Calculus.
Nature of the process of differentiation. Differentiation of , , , and and of easy rational functions of a function.
Second differentiation. Velocities and rates of increase and diminution in
general. Maxims and minima of simple functions of one variable. Simple cases of
approximation, e.g. calculation of small correction. Integration as converse of
differentiation. Definite and indefinite integrals of , , , . Easy
applications of the Calculus. Tangents and normal to curves, simple areas and
volumes.
4.
Higher Plane and Solid
Geometry.
Including the more advanced geometry of the triangle and its
associated circles, and the elementary geometry of the plane and sphere, and
the simpler regular solids.
5.
Coordinate Geometry.
Straight line and circle.
6.
Conics.
Coordinate geometry of the parabola, ellipse, and hyperbola.
7.
Applied Mathematics. (i.e.
Dynamics, including Kinematics, Statics and Kinetics).
Candidates will be expected to solve problems involving use of the
Diff. and Int. Calculus.
One of the above divisions is sufficient
for a pass, no Candidate may take more than two divisions.
（中文參考譯文）
高等數學（B組：選修科目）
1.
進階代數
對課程2(b)[即Mathematics一科Algebra部分]以及以下內容有更全面的了解：三個未知量的聯立方程式。 二次方程式理論；虛數和複數。 排列和組合；二項式定理；部分分式；指數和對數級數。
2.
進階三角學
三角函數的週期性；反三角函數；恒等式與轉換；多多倍函數的展式；三角函數和角度的角量度之間的關係；三角形的解；三角形和四邊形的性質；棣美弗定理及其應用；消去法。
3.
初等微積分
微分過程的本質。、、、和簡單有理函數的微分。二階微分。速度及一般的增加率和減少率。單變數簡單函數的最大值和最小值。簡單的近似個案，例如微增量的計算。積是微分的逆運算。、、、 的定積分和不定積分。微積分的簡單應用。曲線的切線和法線、簡單面積和體積。
包括更高級的三角形及其相關圓的幾何形狀，以及平面和球體的基本幾何形狀，以及更簡單的正多面體。
5.
坐標幾何
直線和圓。
6.
圓錐曲線
拋物線、橢圓和雙曲線的座標幾何。
7.
應用數學。（即動力學，包括運動學、靜力學和動力學）
Mechanics (Group B: Optional Subjects)
Elementary notion of displacement, Velocity
and Acceleration. Motion of a body with constant Acceleration. Resolution and
Composition of Velocities, Accelerations, etc. Elementary notions of Mass and Momentum.
Elementary notions of Force as measured by rate of change of Momentum. Newton’s
Laws of Motion. Kinetic Energy, and Work. Units of Force, and Measurement.
Balancing of Forces. Torques or Moments. Conditions for the equilibrium of
Three Parallel Forces. Resolution and Composition of Parallel Forces in one
Plane. Centre of Parallel Forces. Centre of Gravity. Stable, Unstable, and
Neutral Equilibrium. Conditions for the equilibrium of Three Forces not
parallel. Triangle and Parallelogram of Forces. Moments Simple illustrations of
Conditions of Equilibrium and of the Principle of Work, as in levers, pulleys,
the inclined plane, etc. Pressure in Liquids; variations with depth.
Transmission of Liquid Pressure; Hydraulic Press. Pressure on immersed and
floating bodies. Density; methods of determining Relative Densities. Relation
between volume and pressure in Gases. Atmospheric Pressure. (1933 cancel
Atmospheric Pressure)
（中文參考譯文）
力學（B 組：選修科目）
位移、速度和加速度的基本概念。物體以恆定加速度運動。速度、加速度等的分析與合成。質量與動量的基本概念。透過動量變化率測量的力的基本概念。牛頓運動定律。動能和作功。力的單位和測量。力的平衡。扭力或力矩。三個平行力平衡的條件。一個平面內平行力的分解與合成。平行力中心。重心。穩定、不穩定和中性平衡。三力平衡的條件不平行。三角形和平行四邊形的力。力矩。平衡條件和工作原理的簡單說明，如槓桿、滑輪、斜面等。隨深度的變化。液體壓力傳輸；液壓機。浸入水中和漂浮物體上的壓力。密度；測定相對密度的方法。氣體體積和壓力之間的關係。大氣壓力。（1933年取消大氣壓力）
附錄3：初級試（Junior Local Examination）1915-1932
1922-1924, 1925
Arithmetic
General:
Excluding, however, circulating decimals, extraction of the cube root,
discount, scales of notation, compound interest, stocks and shares, practices.
A knowledge of the tables of weights and measures in common use and of the
metric system is essential. Algebraic symbols may be employed, but all the
steps of the reasoning must be clearly explained.
（中文參考譯文）
算術
一般：但是，不包括循環小數、立方根提取、折扣、記數法、複利息、股票和股份、慣例。了解常用的重量和尺寸表以及公制系統是至關重要的。可以使用代數符號，但必須清楚地解釋推理的所有步驟。
Mathematics
1.
Geometry: As represented by the
subject-matter of Schedule A(1) and B(1). These use of algebraic symbols is
permitted. (1922-1924)
Geometry: The papers in
Geometry will contain questions on both Theoretical and Practical Geometry.
No text-book is specified,
and none need be quoted in answering questions, but the requirements are
covered by Hall and Stevens “School Geometry”, Part I. (1925, 1927)
2.
Elementary Algebra: Elementary
algebraical processes; symbolical representation; fractions; factors; square
root; simple equations containing one or more unknown quantities; quadratic
equations containing one unknown quantity and problems leading thereto. Graphic
solution of equations. Simple questions (not requiring theoretical proofs) on
fractional and negative indices.
3.
Advanced Algebra: The solution
of easy quadratic equation in two unknown quantities; HCF and LCM; Indices;
Surds; Ratio and Proportion; the elementary properties and use of Logarithms;
Simple Identities; Graphic solution of simple simultaneous equations.
(1922-1924)
Advanced Algebra: The
solution of easy quadratic equation in two unknown quantities; HCF and LCM;
Indices; Surds; Ratio and Proportion; the elementary properties and use of
Logarithms; Simple Identities; Graphic solution of simple simultaneous
equations; the Progressions. (1925, 1927)
4.
Mensuration: The relations
between the linear dimensions and the areas or volumes of a triangle,
parallelogram, circle, prism, cylinder, cone, sphere; heights and distances,
including elementary cases in three dimensional the elements of land-surveying.
A knowledge of the use of squared paper and of the use of logarithms in
numerical calculations will be assumed.
5.
Plane Trigonometry: Measurement
of angles in degrees, minutes, and seconds; definition and simple relations of
the trigonometrical ratios of positive angles, and their numerical values in
simple cases; graphic representation of the trigonometrical functions; simple
relations between the side and angles of a triangle; use of tables of natural
sines and tangents; solution of right-angled triangles; easy problems requiring
the use of trigonometry.
Candidates
can pass in Mathematics by qualifying in two division, one of which must be
either (a) or (b), provided that no candidate may pass by taking (b) and (c)
only. No candidate can obtain distinction who does not reach a sufficient
standard in at least three of the divisions (a) to (e).
（中文參考譯文）
數學
1.
幾何：如附表 A(1) 和 B(1) 的主題所示。
允許使用代數符號。（1922-1924）
幾何：幾何試卷將包含理論幾何和實用幾何的問題。
沒有指定教科書，也不需要在回答問題時引用任何教科書，但Hall及Stevens的《School Geometry Part I》涵蓋了這些要求。（1925，1927）
2.
初等代數：基本代數過程；符號表示；分數；因數；平方根；包含一個或多個未知量的簡單方程式；包含一個未知量的二次方程式及其所導致的問題。方程式的圖形解。關於分數和負指數的簡單問題（不需要理論證明）。
3.
高等代數：兩個未知量的簡單二次方程式的解；最大公因數和最大公因數；指數；根式；比和比例；對數的基本特性和使用；簡單恒等式；簡單聯立方程式的圖形解。（1922-1924）
兩個未知量的簡單二次方程式的解；最大公因數和最大公因數；指數；根式；比和比例；對數的基本特性和使用；簡單恒等式；簡單聯立方程式的圖形解；級數。（1925、1927）
4.
測量：三角形、平行四邊形、圓形、稜柱、圓柱體、圓錐體、球體的線性尺寸與面積或體積的關係；高度和距離，包括三維土地測量要素的基本情況。假設您了解方格紙的使用以及數值計算中對數的使用。
5.
平面三角學：以度、分、秒為單位測量角度；正角三角比的定義和簡單關係，以及簡單情況下的數值；三角函數的圖像表示；三角形的邊和角之間的簡單關係；使用自然正弦和正切表；直角三角形的解；需要使用三角學的簡單問題。
考生可以通過兩個級別的資格考試來通過數學，其中一個級別必須是 (a) 或 (b)[即幾何或初等代數]，但任何考生都不能僅通過 (b) 和 (c) 來通過[即初等代數及高等代數]。 如果候選人在 (a) 至 (e) 至少三個類別中未達到足夠的標準，則無法獲得榮譽（Distinction）。
Mechanics
該科課程大綱與高級考試相同，唯考基本論點。
附錄4：高級試（Senior Local Examination）1915-1933
1922-1924, 1925
Arithmetic (1925, 1927 need to pass this subject)
General-Questions
may be set in which contracted methods of multiplication of division must be
employed. Algebraic solutions may be employed, but all the steps of the
reasoning must be clearly explained.
（中文參考譯文）
算術（1925、1927規定需於此科及格）
一般－可能會提出必須使用乘除法的簡化方法的問題。可以使用代數解，但必須清楚解釋推理的所有步驟。
Mathematics
1.
Algebra: This includes the
course prescribed in Elementary Algebra and Advanced Algebra for the Junior
Examination.
2.
Geometry: This paper in
Geometry will contain questions on both Theoretical and Practical Geometry.
As represented by the
subject-matter of Schedules A(1), A(2), B(1) and B(2). (1922-1924)
Geometry: The papers in
Geometry will contain questions on both Theoretical and Practical Geometry.
No textbook is specified,
and more need be quoted in answering questions, but the requirements are
covered by Hall and Stevens “School Geometry” Part I, II and III.
（中文參考譯文）
數學
1.
代數：包括初級試中《初級代數》和《高級代數》規定的課程。
2.
幾何：幾何論文將包含理論幾何與實用幾何的問題。
如附表 A(i)、A(ii)、B(i) 和 B(ii) 的主題所示。（1922-1924）
幾何試卷將包含理論幾何和實用幾何的問題。
沒有指定教科書，也不需要在回答問題時引用任何教科書，但Hall和Stevens之《School Geometry Part I, II, III》涵蓋了這些要求。（1925、1927）
Trigonometry
(1922-1924)
Measurement
of Angles in Degrees, Minutes and Seconds. Definitions and Simple Relations of
the Trigonometrical Ratios of a Positive Angle not greater than two right
angles. Simple Relations between the Sides and Angles of a Triangle. Use of
four-figure logarithmic tables in Simple Problems including the Solution of
Triangles.
(1925,
1927)
Measurement
of Angles in Degrees, Minutes, Seconds, and Radians. Definitions and Simple
Relations of the Trigonometrical Ratios of a Positive Angle not greater than
two right angles. Simple Relations between the Sides and Angles of a Triangle.
Use of four-figure logarithmic tables in Simple Problems including the Solution
of Triangles. Easy trigonometrical equations.
（中文參考譯文）
三角學
（1922-1924）
以度、分和秒為單位的角度測量。不大於兩個直角的正角的三角比的定義和簡單關係。三角形的邊和角之間的簡單關係。在簡單問題中使用四位對數表，包括解三角形。
（1925、1927）
以度、分、秒和弧度為單位的角度測量。不大於兩個直角的正角的三角比的定義和簡單關係。三角形的邊和角之間的簡單關係。在簡單問題中使用四位對數表，包括解三角形。簡易三角方程。
Mechanics
Elementary
notion of displacement, Velocity and Acceleration. Motion of a body with
constant Acceleration. Resolution and Composition of Velocities, Accelerations,
etc. Elementary notions of Mass and Momentum. Elementary notions of Force as
measured by rate of change of Momentum. Newton’s Laws of Motion. Kinetic
Energy, and Work. Units of Force, and Measurement. Balancing of Forces. Torques
or Moments. Conditions for the equilibrium of Three Parallel Forces. Resolution
and Composition of Parallel Forces in one Plane. Centre of Parallel Forces.
Centre of Gravity. Stable, Unstable, and Neutral Equilibrium. Conditions for
the equilibrium of Three Forces not parallel. Triangle and Parallelogram of Forces.
Moments. Simple illustrations of Conditions of Equilibrium and of the Principle
of Work, as in levers, pulleys, the inclined plane, etc. Pressure in Liquids;
variations with depth. Transmission of Liquid Pressure; Hydraulic Press.
Pressure on immersed and floating bodies. Density; methods of determining
Relative Densities. Relation between volume and pressure in Gases. Atmospheric
Pressure.
（中文參考譯文）
位移、速度和加速度的基本概念。物體以恆定加速度運動。速度、加速度等的分析與合成。質量與動量的基本概念。透過動量變化率測量的力的基本概念。牛頓運動定律。動能和作功。力的單位和測量。力的平衡。扭力或力矩。三個平行力平衡的條件。一個平面內平行力的分解與合成。平行力中心。重心。穩定、不穩定和中性平衡。三力平衡的條件不平行。三角形和平行四邊形的力。力矩。平衡條件和工作原理的簡單說明，如槓桿、滑輪、斜面等。隨深度的變化。液體壓力傳輸；液壓機。浸入水中和漂浮物體上的壓力。密度；測定相對密度的方法。氣體體積和壓力之間的關係。大氣壓力。
Higher Mathematics
E.
Higher Geometry (1925 cancelled
this part)
(included the Elementary
Geometry of the plane and sphere and the Algebraical Geometry of the straight
line, circle and conic section).
F.
Higher Algebra and Higher Plane
Trigonometry (1922-1924, 1925 divided two parts)
Higher Algebra
(1922-1924)
A more thorough knowledge
of the course 11(a), together with the following: Simultaneous Equations in
three unknown quantities. Factorization; Imaginary and Complex Quantities; The
Progressions; Permutations and Combinations, the Binomial Theorem for a
positive integral exponent; Partial Fractions; Elimination.
(1925, 1927)
A more thorough knowledge
of the course 11(a) together with the following: Simultaneous Equation in three
unknown quantities. Theory of Quadratic Equations; Imaginary and Complex
Quantities. Permutations and Combinations; the Binomial Theorem; Partial Fractions;
Exponential and Logarithmic Series.
Higher Trigonometry
(1922-1924)
The periodicity of circular
functions; inverse circular functions; identities and transformations;
expansions of functions of multiple angle; relations between the circular
functions and the circular measure of an angle; the solution of triangles;
properties of triangles and quadrilaterals; De Moivre’s Theorem and its
applications.
(1925, 1927)
The periodicity of circular
functions; inverse circular functions; identities and transformations;
expansions of functions of multiple angle; relations between the circular
functions and the circular measure of an angle; the solution of triangles;
properties of triangles and quadrilaterals; De Moivre’s Theorem and its
applications; Elimination.
G.
Elementary Differential
Calculus.
(1922-1924)
Nature of the process of
differentiations. Differentiation of , , and of
easy rational functions. Rules for differentiation of sums, products,
quotients, functions of a function. Second differentiation. Tangents and normal
of simple curves. Velocities, including rates of increase and diminution in
general. Maxima and minima of simple functions of one variable. Simple cases of
approximation, e.g. calculation of small corrections. Acceleration in simple
cases.
(1925, 1927)
Nature of the process of
differentiation. Differentiation of x^n, sin x, cos x, and tan x and of easy
rational functions of a function. Second differentiation. Velocities and rates
of increase and diminution in general. Maxima and minima of simple functions of
one variable. Simple cases of approximation, e.g. calculation of small
corrections. Integration as converses of differentiation. Definite and
indefinite of integrals of x^n, sin x, cos x, sec^2x. Easy applications of the
Calculus. Tangents and normal to curves, simple areas and volumes.
H.
Applied Mathematics (Statics
and Dynamics).
Opportunity will be given
for Candidates to show knowledge of methods involving the use of the
Differential and Integral Calculus.
(1925, 1927)
Applied Mathematics. (i.e.
Dynamics, including Kinematics, Statics and Kinetics). Candidates will be
expected to solve problems involving use of the Diff. and Int. Calculus.
I.
Higher Plane and Solid
Geometry. (1925 added)
Including the more advanced
geometry of the triangle and its associated circles, and the elementary
geometry of the plane and sphere, and the simpler regular solids.
J.
Coordinate Geometry. (1925
added)
Straight line and circle.
K.
Conics. (1925 added)
Coordinate geometry of the
parabola, ellipse, and hyperbola.
One
of the above divisions is sufficient for a pass, no Candidate may take more
than two divisions.
（中文參考譯文）
高等數學
A.
進階幾何（1925年取消了這一部分）
（包括平面、球面的初等幾何和直線、圓、圓錐曲線的代數幾何）。
B.
進階代數和進階平面三角學（1922-1924、1925年分成兩部分）
進階代數
（1922-1924）
更全面地了解課程
11(a)，以及以下內容：三個未知量的聯立方程式。因式分解；虛數和複數；數列；排列與組合、正整指數的二項式定理；部分分式；消去法。
（1925、1927）
對課程11(a)以及以下內容有更全面的了解：三個未知量的聯立方程式。 二次方程式理論；虛數和複數。
排列和組合；二項式定理；部分分式；指數和對數級數。
進階三角學
（1922-1924）
三角函數的週期性；反三角函數；恒等式與轉換；多倍函數的展式；三角函數和角度的角量度之間的關係；三角形的解；三角形和四邊形的性質；棣美弗定理及其應用。
（1925、1927）
三角函數的週期性；反三角函數；恒等式與轉換；多倍函數的展式；三角函數和角度的角量度之間的關係；三角形的解；三角形和四邊形的性質；棣美弗定理及其應用；消去法。
C.
初等微積分
（1922-1924）
微分過程的本質。、、 和簡單有理函數的微分。和、積、商、函數的函數的微分法則。二階微分。簡單曲線的切線和法線。速度，包括一般的增加率和減少率。
單變數簡單函數的最大值和最小值。簡單的近似個案，例如
微增量的計算。簡單情況下的加速度。
（1925、1927）
微分過程的本質。、、、和簡單有理函數的微分。二階微分。速度及一般的增加率和減少率。單變數簡單函數的最大值和最小值。簡單的近似個案，例如微增量的計算。積是微分的逆運算。、、、 的定積分和不定積分。微積分的簡單應用。曲線的切線和法線、簡單面積和體積。
D.
應用數學（靜力學和動力學）
考生將有機會展示涉及使用微積分和積分的方法的知識。
（1925、1927）
應用數學。（即動力學，包括運動學、靜力學和動力學）考生需要解決涉及使用微分和積分的問題。
E.
進階平面和立體幾何（1925年新增）
包括更高級的三角形及其相關圓的幾何形狀，以及平面和球體的基本幾何形狀，以及更簡單的正多面體。
F.
坐標幾何（1925年新增）
直線和圓。
G.
圓錐曲線（1925年新增）
拋物線、橢圓和雙曲線的座標幾何。
上述一個部分足以通過，任何考生不得超過兩個部分。
* General Instructions for Geometry (for 1922-1924)
The
papers in Geometry will contain questions on Practical and Theoretical
Geometry. Every Candidate will be expected to answer questions in both branches
of the subject.
The
questions on Practical Geometry will be set on the constructions contained in
the annexed Schedule A, together with easy extensions of them. In cases where
the validity of a construction is not obvious, the reasoning by which it is
justified may be required. Every Candidates must provide himself with a ruler
graduated in inches and tenths of an inch, and in centimetres and milimetres, a
set square, a protractor, compasses, and a fairly hard pencil. All figures must
be drawn accurately and distinctly. Questions may be set in which the use of
the set square or of the protractor is forbidden.
The
questions on Theoretical Geometry will consist of theorems contained in the
annexed Schedule B, together with questions upon these theorems, easy
deductions from them, and arithmetical illustrations, Any proof of a
proposition will be accepted which appears to the examiners to form part of a
systematic treatment of the subject; the order in which the theorems are stated
in Schedule B is not imposed as a sequence of their treatment. In the proof of
theorems and deductions from them, the use of hypothetical constructions will
be permitted.
Schedule A – Practical
A(i)
8.
Bisection of angles and of
straight lines.
9.
Construction of perpendiculars
to straight lines.
10.
Construction of parallels to a
given straight line.
11.
Simple cases of the
construction of triangles from given data.
12.
Division of straight lines into
a given number of equal parts or into parts in any given proportions.
13.
Construction of tangents to a
circle and of common tangents to two circles.
14.
Construction of circumscribed,
inscribed and escribed circles of a triangle.
A(ii)
8.
Simple cases of the
construction of circles from sufficient data.
9.
Construction of regular figures
of 3, 4, 6, or 8 sides in or about a given circle.
10.
Construction of a triangle
equal in area to given polygon.
11.
Construction of rectilineal
figures from given data.
12.
Construction of solve
graphically a quadratic equation.
13.
Construction to divide a given
line in medial section.
Schedule B – Theoretical
B(i)
Angles
at a point*
3.
If a straight line stands on
another straight line, the sums of the two angles no formed is equal to two
right angles; and the converse.
4.
If two straight lines
intersect, the vertically opposite angles are equal.
Parallel
Straight Lines
3.
When a straight line cuts two
other straight lines,
(i) a pairs alternate
angles are equal,
or (ii) a pair of
corresponding angles are equal,
or (iii) a pair of interior
angles on the same side of the cutting line are together equal to two right
angles,
then the two straight lines
are parallel; and the converse.
4.
Straight lines which are parallel
to the same straight line are parallel to one another.
Triangles
and Rectilinear Figures
12.
The sum of the angles of a
triangle is equal to two right angles.
13.
If sides of a convex polygon
are produced in order, the sum of the angles so formed is equal to four right
angles.
14.
If two triangles have two sides
of the one equal to two sides of the other, each to each, and also the angles
contained by those sides equal, the triangles are congruent.
15.
If two triangles have two angles
of the one equal to two angles of the other, each to each, and also one side of
the one equal to the corresponding side of the other, the triangles are
congruent.
16.
If two sides of a triangle are
equal, the angles opposite to these sides are equal, and the converse,
17.
If two triangles have the three
sides of the one equal to the three sides of the other, each to each, the
triangles are congruent.
18.
If two right-angled triangles
have their hypotenuses equal, and one sides of the one equal to one side of the
other, the triangles are congruent.
19.
If two sides of a triangle are
unequal, the greater side has the greater angle opposite to it; and the
converse.
20.
Of all the straight lines that
can be drawn to a given straight line from a given point outside it, the
perpendicular is the shortest.
21.
The opposite sides and angles
of a parallelogram are equal, each diagonal bisects the parallelogram, and the
diagonals bisect one another.
22.
If there are three or more
parallel straight lines, and the intercepts made by them on any straight line
that cuts them are equal, then the corresponding intercepts on any other
straight line that cuts them are also equal.
Areas.
5.
Parallelograms on the same or
equal bases and of the same altitude are equal in area.
6.
Triangles on the same or equal
bases and of the same altitude are equal in area.
7.
Equal triangles on the same or
equal bases are of the same altitude.
8.
The square on the hypotenuse of
a right-angled triangle is equal to the sum of the squares on the other two
sides; and the converse.
The
Circle.
10.
A straight line, drawn from the
centre of a circle to bisect a chord which is not a diameter, is at right
angles to the chord; conversely, the perpendicular to a chord from the centre
bisects the chord.
11.
There is one circle, and one
only, which passes through three given points not in a straight line.
12.
Equal chords of a circle are
equidistant from the centre; and the converse.
13.
The tangent at any point of a
circle and the radius through the point are perpendicular to one another.
14.
If two circles touch, the point
of contact lies on the straight line through the centres.
15.
The angle which an are of a
circle subtends at the centre is double that which is subtends at any point on
the remaining part of the circumference.
16.
Angles in the same segment of a
circle are equal; and, if the line joining two points subtends equal angles at
two other points on the same side of it, the four points lie on a circle.
17.
The angle in a semicircle is a
right angle; the angle in a segment greater than a semicircle is less than a
right angle; and the angle is a segment less than a semicircle is greater than
a right angle.
18.
The opposite angles of any
quadrilateral inscribed in a circle are supplementary; and the converse.
B(ii)
7.
Illustrations and explanations
of the geometrical theorems corresponding to the following algebraical
identities:
8.
The square on a side of a
triangle is greater or less than the sum of the squares on the other two sides,
according as the angle contained by those sides is obtuse or acute. The
difference is twice the rectangle contained by one of the two sides and the
projection on it of the other.
9.
In equal circles (or, in the
same circle) (i) if two arcs subtend equal angles at the centres, they are
equal; (ii) conversely, if two arcs are equal, they subtend equal angles at the
centres.*
10.
In equal circles (or, in the
same circle) (i) if two chords are equal, they cut off equal arcs; (ii)
conversely, if two arcs are equal, the chords of the arcs are equal.*
11.
If a straight line touches a
circle, and from the point of contact a chord is drawn, the angles which this
chord makes with the tangent are equal to the angles in the alternate segments.
12.
If two chords of a circle
intersect either inside or outside the circle, the rectangle contained by the
parts of the one is equal to the rectangle contained by the parts of the other.
Loci.
5.
The locus of a point which is
equidistant from two fixed points is the perpendicular bisector of the straight
line joining the two fixed points.
6.
The locus of a point which is
equidistant from two intersecting straight lines consists of the pair of
straight lines which bisect the angles between the two given lines.
7.
The loci of points which move
so that the sum or difference of their distances from two fixed points or from
two given lines are constant.
8.
The construction of triangles
from given data by the methods of the intersection of loci.
*
Proofs of these propositions will not be required.
（中文參考譯文）
*幾何一般說明（1922-1924適用）
幾何試卷將包含實用幾何和理論幾何的問題。 每個考生都需要回答該主題的兩個分支的問題。
實用幾何問題將根據附件附表A中包含的結構以及它們的簡單擴充進行設定。如果解釋的有效性不明顯，則可能需要證明其合理性的推理。每個考生必須為自己準備一把以英吋和十分之一英吋為單位、以公分和毫米為單位的尺、一個三角尺、一個量角器、圓規和一支硬身鉛筆。所有圖形必須準確清晰地繪製。可能會提出禁止使用三角板或量角器的問題。
理論幾何問題將包括附表B中包含的定理，以及關於這些定理的問題、從中進行的簡單推論以及算術插圖。任何在考官看來構成命題一部分的命題證明都將被接受。受試者的系統治療；附表B中陳述定理的順序並未強加為處理它們的順序。在證明定理和推論時，允許使用假設結構。
附表 A –
實用
A(i)
8.
角和直線的平分。
9.
垂直於直線的繪製。
10.
構造與給定直線的平行線。
11.
繪製給定資料構造三角形的簡單情況。
12.
將直線劃分為給定數量的相等部分或按任意給定比例劃分。
13.
繪製圓的切線和兩個圓的公切線。
14.
三角形的外接圓、內切圓和外切圓的繪製。
A(ii)
8.
根據足夠的數據建立圓形的簡單個案。
9.
在給定圓內或圍繞給定圓繪製3、4、6 或 8 條邊的規則圖形。
10.
繪製面積等於給定多邊形的三角形。
11.
根據給定資料建立直線圖形。
12.
繪製以圖形方式求解二次方程式。
13.
在內側部分劃分給定線的繪製。
附表 B –
理論
B(i)
直線上的角*
3.
若一條直線與另一條直線重疊，則兩條直線所成的角和等於兩個直角；
反之亦然。[直線上的鄰角]
4.
如果兩條直線相交，則垂直之對角相等。[對頂角]
平行直線
3.
當一條直線與另外兩條直線相交時，
(i) 一對内錯角相等，
或 (ii) 一對同位角相等，
(iii) 截線同一側的一對內角合在一起等於兩個直角，
那麼兩條直線平行；
反之亦然。[内錯角相等、同位角相等、同旁内角互補]
4.
與同一條直線平行的直線彼此平行。
三角形和直線圖形
12.
三角形的內角和等於兩個直角。[三角形內角和]
13.
如果依序產生凸多邊形的邊，則所形成的角和等於四個直角。[多邊形外角和]
14.
如果兩個三角形的一邊與另一個三角形的兩邊相等，而這些邊所含的角也相等，則這兩個三角形全等。[SAS]
15.
如果一個三角形的兩個角等於另一個三角和的兩個角，且一個三角形的一邊等於另一個三角形的對應邊，則這兩個三角形全等。[ASA、AAS]
16.
如果三角形的兩邊相等，則與這些邊相對的角也相等，反之亦然，
17.
如果一個三角形的三邊等於另一個三角形的三邊，則這兩個三角形全等。[SSS]
18.
如果兩個直角三角形的斜邊相等，且直角三角形的一側等於另一個直角三角形的一側，則這兩個三角形全等。[RHS]
19.
若三角形的兩邊不相等，則較大邊的所對的角較大；反之亦然。[大邊對大角、大角對大邊]
20.
在從給定直線外部的給定點可以繪製到給定直線的所有直線中，垂線是最短的。
21.
平行四邊形的對邊和角相等，每條對角線平分平行四邊形，對角線互相平分。[平行四邊形對角、平行四邊形對邊、平行四邊形對角線]
22.
如果有3條或3條以上平行直線，且它們在與它們相交的任何一條直線上的截距相等，則在與它們相交的任何其他直線上相應的截距也相等。[截線定理]
面積
5.
相同或相等底且相同高度的平行四邊形的面積相等。
6.
相同或相等底、相同高的三角形面積相等。
7.
相同或相等的底上的相等三角形具有相同的高度。
8.
直角三角形斜邊的平方等於另外兩邊的平方和；反之亦然。[畢氏定理]
圓
10.
由圓心平分非直徑弦的直線與弦成直角；相反，從中心到弦的垂線平分該弦。[圓心至弦的垂線平分弦、圓心至弦中點的連線垂直弦]
11.
有一個圓，而且只有一個圓，它穿過三個給定的點，但不在一條直線上。
12.
圓的等弦距圓心等距；反之亦然。[等弦對等弦心距、等弦心距對等弦]
13.
圓上任一點的切線和經過該點的半徑互相垂直。[切線⊥半徑]
14.
如果兩個圓相接觸，則接觸點位於通過圓心的直線上。
15.
圓的圓心所對的角度是圓周其餘部分任意點所對的角度的兩倍。[圓心角兩倍於圓周角]
16.
圓內同一弓形的角相等；並且，如果連接兩點的線與同一側的另外兩點所成的角度相等，則這四個點位於一個圓上。[同弓形內的圓周角、同弓形內的圓周角的逆定理]
17.
半圓內的角是直角；
大於半圓的線段內的角小於直角；
角度是小於半圓大於直角的線段。[半圓上的圓周角]
18.
圓內切四邊形的對角互補；反之亦然。[圓內接四邊形對角、對角互補]
B(ii)
7.
對應於以下代數恆等式的幾何定理的圖示與解釋：
[乘法分配律、完全平方、平方差]
8.
三角形的一邊的平方大於或小於另外兩邊的平方和，取決於這些邊所含的角度是鈍角還是銳角。
其差值是兩邊之一和另一邊在其上的投影所包含的矩形的兩倍。
9.
在等圓中（或在同一個圓中） (i) 如果兩個圓弧的圓心所對的角度相等，則它們相等； (ii) 相反，如果兩條弧相等，則它們在中心處所對的角度相等。*[等角對等弧、等弧對等角]
10.
在等圓中（或在同一個圓中） (i) 如果兩個弦相等，則它們截出的弧相等；
(ii) 相反，若兩條弧相等，則這兩條弧的弦也相等。*[等弦對等弧、等弧對等弦]
11.
如果一條直線與圓相切，並從接觸點畫一條弦，則該弦與切線所成的角度等於交替線段中的角度。[交錯弓形上的圓周角]
12.
如果圓的兩條弦在圓的內部或外部相交，則其中一個的部分所包含的矩形等於另一個的部分所包含的矩形。
軌跡
5.
與兩個固定點等距的點的軌跡是連接兩個固定點的直線的垂直平分線。
6.
與兩條相交直線等距的點的軌跡由平分兩條給定線之間的角度的一對直線組成。
7.
移動的點的軌跡，使得它們距離兩個固定點或兩條給定線的距離總和或差恆定。
8.
透過軌跡交集的方法根據給定資料建立三角形。*
*不需要這些命題的證明。
附錄5：全港中學畢業會考（Hong Kong University School Certificate Examination）1935-1937
1934,1935,1936-1937,1937-1938
（未能得知此時期的課程内容）
附錄6：港大入學試（Matriculation Examination）1938-1953
Version 1: 1938-1939,
1939-1940, 1940, 1941, 1941-1942
Elementary Mathematics (Group D)
Arithmetic, Algebra, Geometry, Trigonometry
算術、代數、幾何、三角學
（未能得知此時期的課程内容）
Advanced Mathematics (Group E) / Higher
Mathematics (1941 and after)
（未能得知此時期的課程内容）
Version 2: 1948-1950
（未能得知此時期的課程内容）
Version 3: 1951-1952,
1952-1953
Matriculated students will not be admitted
to courses at the University of Hong Kong in the Faculties of Engineering or
Science, or to courses in the Department of Mathematics in the Faculty of Arts,
if they have offered the Lower Mathematics aper in the Matriculation
Examination; nor will they be admitted to courses in the Department of Chinese
in the Faculty of Arts, unless they have offered the ordinary Chinese paper as
their second language in the Matriculation Examination.
Elementary Mathematics
The examination will consist of two
two-hour papers
To pass, candidates must satisfy the
examiner in both of these two papers
A. Algebra with Arithmetic
Algebra.
Ø Elementary algebraic processes,
Ø symbolic representation,
Ø factors,
Ø fractions,
Ø square roots,
Ø simple and quadratic equations in one or more unknown quantities,
Ø indices,
Ø surds,
Ø ratio and proportion,
Ø HCF and LCM,
Ø Logarithms,
Ø identities,
Ø remainder theorem,
Ø graph and their interpretation,
Ø progressions,
Ø problems.
Arithmetic.
Ø General, but excluding cube roots and scales of notation.
Ø Algebraic symbols and contract methods may be used.
B. Geometry with Trigonometry
Geometry.
Ø The general geometry of the triangle and its associated circles.
Ø Similar triangles and polygons.
Ø Simple loci.
Ø Questions will be set on both practical and theoretical geometry,
with emphasis on deductions.
Ø No text book is
specified, and none need be quoted in the answers, but the requirements are
covered by Hall & Stevens “School Geometry” Part I to V.
Trigonometry.
1.
Measurement of angles in
degrees, minutes, seconds and in radians.
Trigonometrical ratios of positive and negative angles.
Graphical representation of direct trigonometrical functions.
2.
Relations between the
trigonometrical ratios of (1) the same angle, (2) an angle and its complement
of its supplement.
Sum, difference, and product formulae for two angles.
Ratios for 2A, 3A and 1/2A.
3.
Simple identities.
Easy trigonometric equations.
Relations between the sides and angles of a triangle.
Circum-centre, in-centre-ex-centre, ortho-centre.
Formulae for area of a triangle.
Solution of triangles.
4.
Use of mathematical tables.
5.
Problems.
Lower Mathematics
1.
As an alternative to Elementary
Mathematics, Lower Mathematics consists of two two-hour papers
2.
To pass, candidates must
satisfy the examiner in both of these papers
3.
The syllabus will be same as
for Elementary Mathematics (excluding Trigonometry), but questions will be set
more on the elementary part of the syllabus, and will be easier and more
straightforward.
（中文參考譯文）
已入學的學生如在入學考試中取得了初級數學成績，將不會被錄取入讀香港大學工程學院或理學院的課程，或文學院數學系的課程；除非在入學考試中以初級中文試卷作為第二語言，否則不得進入文學院中文系課程。
基本數學
考試將包括兩張兩小時的試卷
要通過，考生必須在這兩篇考卷中都滿足考官的要求
A. 代數與算術
代數
Ø 基本代數過程，
Ø 符號表示，
Ø 因數，
Ø 分數，
Ø 平方根，
Ø 一個或多個未知量的簡單二次方程，
Ø 指數，
Ø 根式，
Ø 比和比例，
Ø 最大公因數和最小公倍數，
Ø 對數，
Ø 恆等式，
Ø 餘式定理，
Ø 圖像及其解釋，
Ø 數列，
Ø 問題。
算術
Ø 一般，但不包括立方根和符號尺度。
Ø 可以使用代數符號和合規方法。
B. 幾何與三角學
幾何
Ø 三角形及其相關圓的一般幾何形狀。
Ø 相似的三角形和多邊形。
Ø 簡單軌跡。
Ø 問題將圍繞實踐和理論幾何，重點是演繹。
Ø 沒有指定教科書，也不需要在答案中引用任何教科書，但要求已在Hall及Stevens之《School Geometry Part I to
V》中涵蓋。
三角學
1.
以度、分、秒和弧度為單位測量角度。
正角和負角的三角比。
直接三角函數的圖形表示。
2.
(1)同角、(2)角及其補角的三角比之間的關係。
兩個角度的和、差和乘積公式。
2A、3A 和 1/2A 的三角比。
3.
簡單恒等式。
簡單三角方程。
三角形的邊和角之間的關係。
外心、内心、形心、重心。
三角形面積公式。
三角形的解。
4.
數表的使用。
5.
問題。
初級數學
1.
作為基本數學的替代，初級數學由兩份兩小時的試卷組成
2.
要通過，考生必須在這兩份考卷中都滿足考官的要求
3.
課程大綱與基本數學（不含三角函數）相同，但題目將更集中在課程大綱的初等部分，並且會更簡單、更直接。