Euclid's geometry is held by some to achieve the ideal of an aprioristic science.
An aprioristic science is supposed to fulfill the Kantian aspiration of knowledge attained by synthetic propositions a priori: statements that by virtue of their logical content describe or explain the empirical world accurately and definitely.
Apriorism means that man is capable of thinking up or discovering in his mind first principles from which a true and complete science of (aspects of) the real world, geometry or economics e.g., can be derived.
Aprioristic science is the ultimate dream of the rationalist come true.
It is the ultimate dream of those come true who cannot accept insight as valuable, unless that insight is held to be absolute, indubitably certain, protected from refutation and revision.
Coming up with a science in this vein is what the Latin phrase more geometrico refers to.
In the manner of (Euclidean) geometry.
The problem is: Euclid's geometry does not do the trick.
And, I am afraid - in fact, I am glad -, no other (attempt at) science does.
The inconsistencies of Euclid's geometry that the gentleman in the below video talks about, have been successfully rectified by David Hilbert, who transformed Euclidean geometry into an axiomatically consistent theory.
However, Hilbert renounced any ambition to declare his axioms to be true, let alone indubitably true, and to derive from them propositions of factual truth, of empirical validity, let alone of ultimate veracity concerning the real world. Hilbert's sole ambition was to achieve a logically consistent theory. Period. That's a far cry from the triumph of synthetic statements a priori that the rationalist so yearns for.
Why am I applauding the failure of the aprioristic programme?
Not only does the aprioristic approach fail to grasp the true power of human cognition, which lies in its wonderful imperfection, it is authoritarian, sophocratic, one big temptation to usurp power on the pretext of absolutely superior knowledge.
If I proved to be wrong on this question, I would feel deeply wounded - as a lover of liberty.