1
Overview (Divine Proportions)
1.
1 Introducing quadrance and spread
Trigonometry is the measurement of triangles.
In classical trigonometry, measurement
uses distance and angle, while in rational trigonometry measurement uses quadrance
and spread....
More

1
Overview (Divine Proportions)
1.
1 Introducing quadrance and spread
Trigonometry is the measurement of triangles.
In classical trigonometry, measurement
uses distance and angle, while in rational trigonometry measurement uses quadrance
and spread.
To appreciate the diﬀerence, let’s consider a speciﬁc triangle in the
decimal number plane from both points of view.
Classical measurements
The triangle with side lengths
7
4
5
A1
1 2
3
A3
A2
q q
q
d1 = 4 d2 = 7 d3 = 5
has respective angles (approximately)
θ1 ≈ 33.
92◦
θ2 ≈ 102.
44◦
θ3 ≈ 43.
64◦
.
There are numerous classical relations between
these six quantities, such as the Sums of angles law
θ1 + θ2 + θ3 = 180◦
,
along with the Cosine law, the Sine law, and others.
Typically these laws involve the
trigonometric functions cos θ, sin θ and tan θ, and implicitly their inverse functions
arccos x, arcsin x and arctan x, all of which are diﬃcult to deﬁne precisely without
calculus.
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