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The Egyptian seked
The Egyptians defined the seked as the ratio of the run to the rise, which is the reciprocal of the modern definition of the slope.
Mathematicians of the Greco-Roman world
This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bc) to Hypatia of Alexandria (c. ad 400). Their names—located on the map under their cities of birth—can be clicked to access their biographies.
In the 4th century bc, Menaechmus gave a solution to the problem of doubling the volume of a cube. In particular, he showed that the intersection of any two of the three curves that he constructed (two parabolas and one hyperbola) based on a side (a) of the original cube will produce a line (x) such that the cube produced with it has twice the volume of the original cube.
Sphere with circumscribing cylinder
The volume of a sphere is 4πr3/3, and the volume of the circumscribing cylinder is 2πr3. The surface area of a sphere is 4πr2, and the surface area of the circumscribing cylinder is 6πr2. Hence, any sphere has both two-thirds the volume and two-thirds the surface area of its circumscribing cylinder.
Conic sections
The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle); the parabola (with one branch); and the hyperbola (with two branches).
Conchoid curve
From fixed point P, several lines are drawn. A standard distance (a) is marked along each line from line LN, and the connection of the points creates a conchoid curve.
Angle trisection using a conchoidNicomedes (3rd century
bce) discovered a special curve, known as a conchoid, with which he was able to trisect any acute angle. Given ∠θ, construct a conchoid with its pole at the vertex of the angle (
b) and its directrix (
n) through one side of the angle and perpendicular to the line (
m) containing one of the angle’s sides. Then construct the line (
l) through the intersection (
c) of the directrix and the remaining side of
Angle trisection using a hyperbolaPappus of Alexandria (
c.320) discovered that a hyperbola could be used to trisect an acute angle. Given ∠θ, construct points along one side such that
ba=
ao=
of, and draw the hyperbola with centre at
oand one vertex at
f. Next, construct the line perpendicular to side
basuch that
clies along the other side of ∠θ. Having established the length of
bc, draw the line
adsuch
Angle trisection using a hyperbolaPappus of Alexandria (
c.320) discovered that a hyperbola could be used to trisect an acute angle. Given ∠θ, construct points along one side such that
ba=
ao=
of, and draw the hyperbola with centre at
oand one vertex at
f. Next, construct the line perpendicular to side
basuch that
clies along the other side of ∠θ. Having established the length of
bc, draw the line
adsuch
mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.
In many cultures—under the stimulus of the needs of practical pursuits, such as commerce and agriculture—mathematics has developed