By A. C. M. van Rooij
Whilst contemplating a mathematical theorem one ought not just to grasp easy methods to turn out it but additionally why and no matter if any given stipulations are valuable. All too usually little recognition is paid to to this facet of the idea and in penning this account of the speculation of genuine features the authors wish to rectify issues. they've got placed the classical conception of actual capabilities in a contemporary surroundings and in so doing have made the mathematical reasoning rigorous and explored the speculation in a lot larger intensity than is frequent. the subject material is largely just like that of normal calculus direction and the thoughts used are simple (no topology, degree thought or useful analysis). therefore an individual who's accustomed to straightforward calculus and needs to deepen their wisdom may still learn this.
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Additional resources for A Second Course on Real Functions
Thus, if a straight-line stood on a(nother) straight-line makes angles, it will certainly either make two right-angles, or (angles whose sum is) equal to two right-angles. (Which is) the very thing it was required to show. 35 ΣΤΟΙΧΕΙΩΝ α΄ ιδ΄ Α Γ Ε Β ∆ Ε ν πρός τινι ε θεί κα τ πρ ς α τ σηµεί δύο ε θε αι µ π τ α τ µέρη κείµεναι τ ς φεξ ς γωνίας δυσ ν ρθα ς σας ποι σιν, π ε θείας σονται λλήλαις α ε θε αι. Πρ ς γάρ τινι ε θεί τ ΑΒ κα τ πρ ς α τ σηµεί τ Β δύο ε θε αι α ΒΓ, Β∆ µ π τ α τ µέρη κείµεναι τ ς φεξ ς γωνίας τ ς π ΑΒΓ, ΑΒ∆ δύο ρθα ς σας ποιείτωσαν· λέγω, τι π ε θείας στ τ ΓΒ Β∆.
I say that side AC is also greater than side AB. For if not, AC is certainly either equal to or less than AB. In fact, AC is not equal to AB. For then angle ABC would also have been equal to ACB [Prop. 5]. But it is not. Thus, AC is not equal to AB. Neither, indeed, is AC less than AB. For then angle ABC would also have been less than ACB [Prop. 18]. But it is not. Thus, AC is not less than AB. But it was shown that (AC) is also not equal (to AB). Thus, AC is greater than AB. Thus, for any triangle, the greater angle is subtended by the greater side.
But CEA and AED were also shown (to be) equal to two right-angles. N. 1]. Let AED have been subtracted from both. N. 3]. Similarly, it can be shown that CEB and DEA are also equal. Thus, if two straight-lines cut one another then they make the vertically opposite angles equal to one another. (Which is) the very thing it was required to show. 39 ΣΤΟΙΧΕΙΩΝ α΄ ι ΄ Α Ζ Ε Β ∆ Γ Η Παντ ς τριγώνου µι ς τ ν πλευρ ν προσεκβληθείσης πεναντίον γωνι ν µείζων στίν. κτ ς γωνία κατέρας τ ν ντ ς κα Εστω τρίγωνον τ ΑΒΓ, κα προσεκβεβλήσθω α το µία πλευρ ΒΓ π τ ∆· λ γω, τι κτ ς γωνία π ΑΓ∆ µείζων στ ν κατέρας τ ν ντ ς κα πεναντίον τ ν π ΓΒΑ, ΒΑΓ γωνι ν.
A Second Course on Real Functions by A. C. M. van Rooij