ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ

ᎦᏍᎩᎶ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᏂᏛᎴᏅᏓ ᎯᎠ 1728 Cyclopaedia.

ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ (ᎠᎪᎢ γεωμετρία; geo = ᎡᎶᎯ, metria = ᎠᏟᎶᏍᏗ) ᏚᎴᏅ ᏥᏄᏍᏗ ᎯᎠ ᏠᎨᏏ ᎥᎦᏔᎲᎢ ᎠᏂᎾᏕᎬ ᎬᏙᏗ spatial ᎠᎾᎵᏐᏈᎸᏍᎬ. ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏥᏄᏍᏛᎩ ᏌᏊ ᎯᎠ ᏔᎵ ᏕᎦᎶᎨᏒ ᎬᏂᎨᏒ ᏄᏍᏛᎢ-ᎪᎯᏊ ᎢᏴ ᏥᎩ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᎯᎠ ᏐᎢ ᎠᎴᏂᏙᎲ ᎯᎠ ᎠᎦᏎᏍᏙᏗ ᏗᏎᏍᏗ. ᎭᏫᎾᏗᏢ ᎪᎯᏊ ᎢᏴ ᏥᎩ ᎢᏧᎳᎪᏗ, geometric ᎠᏓᏅᏖᏗ ᎤᎭ ᏭᏪᏙᎢ generalized ᎦᎸᎳᏗ ᎤᏩᎾᏕᏍᎩ ᎪᏣᎴᏛ ᎠᎴ ᏓᎧᏁᎲ, ᎠᎴ ᎤᎭ ᏭᏪᏙᎢ ᏗᎨᏥᎾᏝᎢ ᎯᎠ ᎢᏗᎬᎾᏗ ᎠᏓᏃᎮᏗ ᎠᎴ ᎠᏓᏓᎶᏙᏗ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᎾᏍᎩ ᎢᎬᏂᏏᏍᎩ Ꮎ ᎤᎪᏗᏗ ᎪᎯᏊ ᎢᏴ ᏥᎩ ᏚᏩᏂᎦᎸ ᎯᎠ ᏠᎨᏏ ᎠᎴ ᎤᏎᎦᏨᎯ ᎦᎾᏄᎪᏨᎢ ᏥᏄᏍᏗ ᎯᎠ ᎠᏂᏁᏉᎬ ᎢᎬᏱ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. (ᎠᎪᏩᏛᏗ ᎡᏍᎦᏂ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎠᎴ ᎠᏓᏃᎮᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ.)

ᎢᎬᏱ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ

ᎯᎠ ᎢᎬᏱᏗᏢᏍᏗ ᎪᏪᎳᏅᎯ ᎠᏓᎴᏂᏍᎬ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏰᎵᏇ ᎾᏍᏋ ᎤᏍᏓᏩᏛᏒ ᎯᎸᎯᏳᎢ Egypt (ᎠᎪᏩᏛᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎭᏫᎾᏗᏢ Egypt), ᎯᎠ ᎯᎸᎯᏳᎢ Indus ᎤᎨᏓᎵᏴ (ᎠᎪᏩᏛᏗ Harappan ᏗᏎᏍᏗ ᎤᎬᏩᎵ), ᎠᎴ ᎯᎸᎯᏳᎢ Babylonia (ᎠᎪᏩᏛᏗ Babylonian ᏗᏎᏍᏗ ᎤᎬᏩᎵ) ᏂᏛᎴᏅᏓ ᏴᎳᏚᏫᏛ 3000 ᎤᏓᎷᎸ. ᎢᎬᏱ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏥᏄᏍᏛᎩ ᎤᏓᏂᏝᏅ ᎠᎦᏛᏗ ᏄᏍᏛ ᎪᎷᏩᏛᏓ ᏚᎵᏍᎪᎵᏴ ᎾᏍᎩ ᎠᏂᏰᎸᏍᎬ ᏂᎦᏅᎯᏒ, ᏓᏍᏓᏅᏅ ᏚᎷᏨ, ᎡᏍᎦᏂ, ᎠᎴ ᏗᎦᏟᏌᏅ ᎪᏪᎵ, ᎦᏙ ᎤᏍᏗ ᎨᏒᎩ ᎤᏙᎷᏩᏛᏓ ᏗᏠᎯᏍᏗ ᎢᎦᏛ ᏗᏙᎳᎩ ᎤᏚᎳᏗ ᎭᏫᎾᏗᏢ ᎠᏟᎶᏍᏗ ᎦᏙᎯ, ᎠᏁᏍᎨᎲ, ᎡᎶᎯ, ᎠᎴ ᏧᏓᎴᏅᏓ ᏥᏳ. ᏄᎾᏛᏅ ᎾᏍᎩ ᎯᎠ ᎨᏒᎩ ᎢᎦᏛ ᏂᎬᎢ ᎦᎶᏄᎮᏛ ᏚᎵᏍᎪᎵᏴ, ᎠᎴ ᎪᎯᏊ ᎢᏴ ᏥᎩ ᎠᏓᏃᎮᏗ ᏄᎵᏂᎬᎬ ᎾᏍᏋ ᎠᏍᏓᏯ ᏩᏗᏱ ᏗᎫᎪᏔᏅ ᎢᎦᏛ ᎠᏂ ᏄᏠᏯᏍᏛᎾ ᎯᎠ ᎬᏙᏗ ᎠᏓᏃᎮᏗ. ᎾᏍᎩᎾᎢ ᏱᏓᏟᎶᏍᏔᏅ, ᎢᏧᎳ ᎯᎠ Egyptians ᎠᎴ ᎯᎠ Babylonians ᎨᏒᎩ ᎦᏯᏙᎲᏍᏗ ᏅᎬᎪᏔᏅᎯ ᎯᎠ ᎣᏩᏒ ᎢᏳᏍᏗ ᎧᏃᎮᏗ ᎬᏩᏚᏫᏛ 1500 ᏧᏕᏘᏴᏓ ᎤᏓᎷᎸ Pythagoras; ᎯᎠ Egyptians ᎠᏰᎲ ᎪᏢᎯᏐᏗ ᏎᏍᏗ ᎾᏍᎩᎾᎢ ᎯᎠ ᎦᏟᏌᏅ ᎪᏪᎵ ᎢᎦᏛ ᏅᎩ ᏧᏅᏏᏯ ᎤᏓᏂᏝᏅ; ᎯᎠ Babylonians ᎠᏰᎲ ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎦᏍᎩᎶ.

ᎯᎸᎯᏳᎢ ᎠᏴᏫᏯ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ (ca. 3000–500 ᎤᏓᎷᎸ)

Harappan ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ

ᎯᎠ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎬᏔᏅᎯ ᎭᏫᎾᏗᏢ ᎯᎠ Indus ᎤᎨᏓᎵᏴ ᎠᏓᏅᏘᏐᏗ ᏧᏴᏢ ᎢᏅᏗᎾ ᎠᎴ Pakistan ᏂᏛᎴᏅᏓ ᏴᏩᏚᏫᏛ 3000 ᎤᏓᎷᎸ ᏥᏄᏍᏛᎩ ᎣᏍᏛ ᏥᏄᏍᏗ ᎠᏓᏌᎳᏙᏗ ᏥᏄᏍᏗ Ꮝ ᏧᏂᏠᏱ ᎭᏫᎾᏗᏢ Egypt ᎠᎴ Mesopotamia, ᎠᎴ ᎾᏍᎩ ᎤᎪᏗᏗ ᎤᏙᎷᏩᏛᏓ ᏥᏄᏍᏗ ᏄᎵᏍᏔᏅ ᎠᏓᏌᎳᏙᏗ ᎦᏁᎸ ᎠᏛᏂᏍᏗᏍᎩ, ᎦᏙ ᎤᏍᏗ ᎨᏒᎢ ᎥᏝ ᏳᏓᎷᎳ ᏂᏛᎴᏅᏓ ᎯᎠ ᎧᎵᏬᎯ grid ᏗᏟᎶᏍᏙᏗ Harappa ᎠᎴ Mohenjo-daro ᎭᏢᏃ ᏕᎦᎳᏅᏛ ᎨᏒᎩ ᎠᏝᏅᎯ ᎠᏥᏄᏉᏫᏍᎬ ᎭᏫᎾᏗᏢ ᎧᎵᏬᎯ ᏚᏳᎪᏛ ᏓᏍᏓᏅᏅ ᏚᎷᏨ. ᎯᎠ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎬᏔᏅᎯ ᎾᎥᎢ ᎪᎯ ᎢᎬᏱ Harappan ᎠᏓᏅᏘᏐᏗ ᏥᏄᏍᏛᎩ ᎾᏍᎩᎾᎢ ᏗᏙᎳᎩ ᎤᏅᏔᏂᏓᏍᏗ, ᎠᎴ ᏥᏄᏍᏛᎩ ᎢᎬᏱᏗᏢ ᎡᎯᏍᏛ ᎬᏙᏗ ᏂᏚᏓᎨᏒ, ᎠᏟᎶᏍᏗ ᎦᏙᎯ ᏄᏓᎨᏒ ᎠᎴ ᏂᎬᎢ ᎠᏓᏌᎳᏙᏗ ᏗᏛᏓᏅᎯ ᎠᏏᎾᏒᎢ, ᎦᏙ ᎤᏍᏗ utilized ᎠᎾᎵᏐᏈᎸᏍᎬ. ᎯᎠ ᎠᎾᎵᏐᏈᎸᏍᎬ ᎾᏍᎩᎾᎢ ᏗᏛᏓᏅᎯ ᎢᎦᎢ 4:2:1 ᎨᏒᎢ ᎢᏧᎳᎭ ᎪᎯ ᎢᎦ ᎠᎦᏎᏍᏔᏅ optimal ᎾᏍᎩᎾᎢ ᎢᏳᎵᏍᏙᏗᏱ ᎠᏚᏓᎸᏙ. ᏗᏛᏓᏅᎯ ᏂᏕᎬᎢ ᎨᏒᎩ ᎭᏫᎾᏗᏢ ᎧᎵᏬᎯ ᎠᎾᎵᏐᏈᎸᏍᎬ 4:2:1. ᏎᏍᏗ ᏂᏚᏓᎨᏒ ᎨᏒᎩ ᏚᎳᏏᏔᏅᎩ ᎾᎿ ᎠᎾᎵᏐᏈᎸᏍᎬ 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, ᎠᎴ 500, ᎬᏙᏗ ᎠᏂᏏᏴᏫᎭ ᏌᏊᎭ ᎦᏛᏍᎬᎢ ᎾᎥᏂᎨᏍᏙᏗᏇ 28 ᏄᏓᎨᏒ, ᎤᏠᏱ ᎯᎠ ᎩᎵᏏ ᏄᏓᎨᏒ ᎠᎴ ᎠᎪᎢ uncia.

ᎤᎪᏗᏗ ᎯᎠ ᏂᏚᏓᎨᏒ uncovered ᎤᎭ ᏭᏪᏙᎢ ᎠᏛᎯᏍᏔᏅ ᎭᏫᎾᏗᏢ ᎾᎿ ᎢᏴ geometrical ᎦᎶᏄᎮᏛ (cuboid, ᏒᏙᏂ, ᏒᏙᏂ, ᎠᎴ ᏒᏙᏂ ᏚᏙᎥ ᎦᏲᎵ) ᎦᏙ ᎤᏍᏗ ᎡᏙᎠ ᎥᎦᏔᎲᎢ ᏄᏦᏍᏛᎾ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎠᏠᏯᏍᏗᏍᎩ ᎯᎠ ᎦᏐᏆᎸ. ᎪᎯ ᎥᎦᏔᎲᎢ ᎾᏍᎩ ᎾᏍᏇ ᎠᏛᎯᏍᏔᏅ ᏗᏟᎶᏍᏗᏍᎩ ᎠᏤᎸ ᎢᎬᏁᏗ ᏚᏳᎪᏛ ᎤᏁᎳᏅ ᎤᏬᏢᏅ ᎠᎴ ᎾᎿᎢ ᎨᏒᎢ ᎪᎯᏳᏙ ᎾᎿ ᎠᏓᏍᏕᎸᏗ Ꮎ ᎾᏍᎩ ᎯᎠ ᏴᏫ ᏰᎵᏇ ᎠᏎᎯᏍᏗ concentric ᎠᎴ ᏗᎦᎾᏗᏫᏍᏗ ᎦᏐᏆᎸ ᎠᎴ ᏦᎢ.

ᎤᏗᏗᏢ ᎯᎠ ᎬᏙᏗ ᎦᏐᏆᎸ ᎭᏫᎾᏗᏢ ᎠᏣᏅᏙ ᎠᏤᎸ ᎢᎬᏁᏗ ᎾᎿᎢ ᎨᏒᎢ ᎢᏧᎳᎭ ᏗᏂᏱᎴᎩ ᎯᎠ ᎬᏙᏗ ᎦᏂᏝᎢ ᎤᏍᏗ ᏓᏆᎴᎷ, ᎯᎠ ᏗᎦᏆᏙᏗ ᎦᏙ ᎤᏍᏗ ᎠᎾᏍᎬᏘ ᎤᎭ ᎠᏰᎲ metallic ᏗᏂᏤᎷᎯᏍᎩ wrapped ᎦᏌᏆᎸ ᎯᎠ ᎠᏍᏛᎢ. ᎢᎦᏛ ᎪᏪᎵᏍᎩ ᎤᏬᎯᏳ ᎪᎯ ᏗᎪᏍᏓᏱ ᎯᎠ ᎪᎱᏍᏗ ᎨᏒ ᎥᎦᏔᎲᎢ ᎯᎠ ᎠᎾᎵᏐᏈᎸᏍᎬ ᎯᎠ ᏂᎦᏅᎯᏒ ᎯᎠ ᎦᏃᎴᏍᎬ ᎯᎠ ᎦᏐᏆᎸ ᎠᎴ Ꮝ ᎠᏰᎵ ᎠᏍᏓᏅᏅ, ᎠᎴ ᎯᎠ ᎢᏴ ᏧᎬᏩᎶᏗᎢ π.

ᎭᏫᎾᏗᏢ Lothal, ᎤᎭᎨᏛ ᎠᎵᏰᏑᏍᏔᏬ-ᎾᏍᎩᏯᎢ ᎤᏯᏍᎦ ᏗᎦᏘᎴᏍᏗ ᎠᏩᏛᏗ ᎬᏙᏗ ᏅᎩ ᎠᎽᏰᎵ ᎠᏂᏏᏴᏫᎭ ᎭᏫᎾᏗᏢ ᏔᎵ ᏧᎬᏩᎶᏗᎢ ᎢᏯᏓᏛᏁᎸ ᏥᏄᏍᏗ ᎠᎳᏂ ᎠᎦᏓᏗ ᎠᏟᎶᏍᏗ ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎾᎿ ᎦᏃᎯᎵᏙ ᎦᏚᎢ ᎠᎴ ᎭᏫᎾᏗᏢ ᎤᏚᎳᏗ ᎭᏫᎾᏗᏢ ᏎᏍᏗ 40–360 ᎢᎦᎢ ᎢᏗᎦᏘ. ᏯᏛᎿ ᎤᏯᏍᎦ ᎪᎱᏍᏗ ᏗᎬᏔᏂᏓᏍᏗ ᎨᏒᎩ ᏄᏓᎷᎸᎾ ᎦᎾᎬᎢ ᎠᏟᎶᏍᏗ 8–12 ᏂᎦᏛ ᎾᎿ ᏕᎨᏒ ᎯᎠ ᎤᏚᎳᏗ ᎠᎴ ᎦᎷᎾᏗ, ᏓᏃᏏᏏᏍᎬ ᎯᎠ ᎠᎽᏰᎵ ᎾᎿ ᎯᎠ ᎡᎳᏗᎨ ᎠᎴ ᎦᎸᎳᏗᏢ ᏧᎬᏩᎶᏗᎢ. ᏗᏕᎶᏆᏍᎩ ᎠᎦᏎᏍᏙᏗ ᎪᎯ ᏥᏄᏍᏗ ᎪᎯᏳᏙ ᎯᎠ Lothal ᎠᏏᎾᏍᏛ ᎠᏰᎲ achieved ᎪᎱᏍᏗ 2,000 ᏧᏕᏘᏴᏓ ᎤᏓᎷᎸ ᎯᎠ ᎠᎪᎢ ᎠᎴ ᎠᏓᏚᏅ ᎬᏙᏗ ᎾᎾᏛᏁᎲ: 8–12 ᎦᏇᏅᏗ ᎠᏰᏗ ᎢᏳᏓᏛ ᎤᏚᎳᏗ ᎠᎴ ᎦᎷᎾᏗ, ᏥᏄᏍᏗ ᎠᏔᎴᏒ ᎠᎹᏱ ᏥᏄᏍᏗ ᎪᎱᏍᏗ ᎬᏔᏂᏓᏍᏗ ᎠᏟᎶᏍᏗ ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎠᎴ ᎠᏎᏱᎩ ᎯᎠ ᎾᎿ ᎦᏙᎬ ᎠᏂᏃᏈᏏ, ᎠᎴ ᎾᏍᎩᎾᎢ ᎠᏂᎩᏍᏗ ᏥᏳ ᏂᏚᏰᎸᏛᎢ. Lothal ᎠᎵᏍᎪᎸᏙᏗ ᏌᏊ ᏦᎢ ᎠᏓᏃᎮᏗ ᏄᏓᎨᏒ Ꮎ ᎠᎴ integrated ᎠᎴ ᎦᏌᏆᎸ (ᎠᏂᏐᎢ ᎠᏩᏛᏗ ᎭᏫᎾᏗᏢ Harappa ᎠᎴ Mohenjodaro). ᎤᏁᎬ ᎪᎳ ᏗᎦᏛᏗ ᏂᏛᎴᏅᏓ Lothal ᎤᎭ ᎯᎠ ᎠᏂᎦᏲᎵ-ᎤᎾᏅᏛ ᏎᏍᏗ ᎠᏰᏗ ᎢᏳᏓᏛ ᎭᏫᎾᏗᏢ Indus ᎠᏓᏅᏘᏐᏗ. ᎯᎠ ᏗᎦᏛᏗ ᎨᏒᎢ 6 mm ᎤᎭᎨᏛ, 15 mm ᎠᏯᏖᎾ ᎠᎴ ᎯᎠ ᏰᎵ ᎠᏩᏛᏗ ᏂᎦᏅᎯᏒ ᎨᏒᎢ 128 mm, ᎠᎴ ᎾᏍᎩ ᎤᏩᏒ 27 ᏓᏂᏍᏆᏗᏍᎬ ᎪᏪᎵ ᎠᎴ ᎬᎪᏩᏛᏗ ᎦᏬᎯᎸᏙᏗ 146 mm, ᎯᎠ ᎾᎿ ᎢᏴᎢ ᎠᏰᎵ ᏓᏂᏍᏆᏗᏍᎬ ᎪᏪᎵ ᎤᎵᏍᏕᎸᏗ ᎠᎴᏂᏙᎲ 1.704 mm (ᎯᎠ ᎤᏍᏗ ᏂᎬᎢ ᎠᏓᏎᎮᏗ ᎬᏙᏗ ᎾᏍᎩᎾᎢ finer ᏂᏚᏰᎸᏛᎢ). ᎯᎠ ᎢᎦᎢ ᏂᎦᏛ ᎦᏟᏌᏅ ᏍᎪᎯ ᏓᏂᏍᏆᏗᏍᎬ ᎪᏪᎵ ᏂᏛᎴᏅᏓ Lothal ᎨᏒᎢ ᎾᎥᏂᎨᏍᏙᏗ ᎯᎠ angula ᎭᏫᎾᏗᏢ ᎯᎠ Arthashastra. ᎯᎠ Lothal craftsmen ᎤᎩᏒᎩ ᎠᎦᏎᏍᏙᏗ ᎠᏍᏓᏩᏛᏍᏗ ᎠᏂᏙᎾᎥ ᎠᎴ ᏚᏳᎪᏛ ᏅᏯ ᏂᏚᏓᎨᏒ ᎾᎥᎢ ᎪᏍᏗᏳᎵ ᎠᏍᏛᎢ ᎤᏓᎷᎸ ᎦᎶᏂᏗ. ᎯᎠ Lothal ᏄᏓᎨᏒ 12.184 gm ᎨᏒᎢ ᎾᎥᏂᎨᏍᏗ ᎢᏗᎦᏗ ᎯᎠ Egyptian Oedet 13.792 gm.

Vedic ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ

ᎾᎯᏳᎨᏒ ᎯᎠ Vedic ᎠᎴᏫᏍᏙᏗ ᎢᏅᏗᎾ ᏗᏎᏍᏗ ᎤᎬᏩᎵ (ca. 1500–500 ᎤᏓᎷᎸ), ᎤᎪᏗᏗ ᏗᎫᎪᏔᏅ ᎠᎴ ᏚᎾᏙᎷᏩᏛᎲ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᎴ ᎠᏩᏛᏗ ᎭᏫᎾᏗᏢ Vedic ᏧᏂᎸᏫᏍᏓᏁᏗ ᏥᏄᏍᏗ ᏄᎵᏍᏔᏅ ᎯᎠ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎧᏁᏨ ᎢᏯᏛᏁᏗ ᎾᏍᎩᎾᎢ ᎯᎠ ᎠᏁᏍᎨᎲ ᏧᏁᎸᏗᏯ altar. ᎾᏍᎩ ᎯᎠ ᎠᏠᏯᏍᏛ ᎯᎠ ᎬᏙᏗ geometric ᎦᎶᏄᎮᏛ, ᎠᏠᏯᏍᏗᏍᎩ ᏦᎢ, ᏓᏍᏓᏅᏅ ᏚᎷᏨ, ᏅᎩ ᏧᏅᏏᏯ, trapezia ᎠᎴ ᎦᏐᏆᎸ, ᎢᏗᎦᏘᎭ ᏗᎬᏩᎶᏒ ᏗᏎᏍᏗ ᎠᎴ ᎡᏍᎦᏂ, squaring ᎯᎠ ᎦᏐᏆᎸ ᎠᎴ vice versa, ᎯᎠ ᎣᏩᏒ ᎢᏳᏍᏗ ᎧᏃᎮᏗ ᎠᎴ ᏙᎪᏩᎸ ᎣᏩᏒ ᏦᎢ ᎪᎷᏩᏛᏓ algebraically, ᎠᎴ ᎠᏓᏃᎮᏗ π (ᎪᏢᎯᏐᏗ 2 ᏎᏍᏗ ᏚᏙᏢᏒᎢ).

ᏥᏄᏍᏗ ᏄᎵᏍᏔᏅ ᎯᎠ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎧᏁᏨ ᎢᏯᏛᏁᏗ ᎾᏍᎩᎾᎢ ᎯᎠ ᎠᏁᏍᎨᎲ ᎾᏍᎩ ᎯᎠ altars, ᎤᎪᏗᏗ ᏗᎫᎪᏔᏅ ᎠᎴ ᏚᎾᏙᎷᏩᏛᎲ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᎴ ᎠᏩᏛᏗ ᎭᏫᎾᏗᏢ Vedic ᏧᏂᎸᏫᏍᏓᏁᏗ. ᎾᏍᎩ ᎯᎠ ᎠᏠᏯᏍᏛ:

• ᎬᏙᏗ geometric ᎦᎶᏄᎮᏛ, ᎠᏠᏯᏍᏗᏍᎩ ᏦᎢ, ᏓᏍᏓᏅᏅ ᏚᎷᏨ, ᏅᎩ ᏧᏅᏏᏯ, trapezia ᎠᎴ ᎦᏐᏆᎸ.
• ᎢᏗᎦᏘᎭ ᏗᎬᏩᎶᏒ ᏗᏎᏍᏗ ᎠᎴ ᎡᏍᎦᏂ.
• Squaring ᎯᎠ ᎦᏐᏆᎸ ᎠᎴ vice versa.
• ᎣᏩᏒ ᏦᎢᎪᎷᏩᏛᏓ algebraically.
• ᏗᎧᏃᎮᎸᎯ ᎯᎠ ᎣᏩᏒ ᎢᏳᏍᏗ ᎧᏃᎮᏗ ᎠᎴ ᎠᏓᏃᎮᏗ ᎪᎯᏳᏙᏗ.
• ᎠᏓᏃᎮᏗ π, ᎬᏙᏗ ᎯᎠ closest ᎠᎴᏂᏙᎲ ᎪᏢᎯᏐᏗ 2 ᏎᏍᏗ ᏚᏙᏢᏒᎢ.

Lagadha (ca. 1350–1200 ᎤᏓᎷᎸ) ᏥᏄᏍᏛᎩ ᏄᏓᎷᎸᎾ ᎯᎠ ᎢᎬᏱᏗᏢᏍᏗ ᎤᎾᏅᏛ ᎠᏓᏃᎮᏗ ᎤᎭ ᎬᏔᏅᎯ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᎴ ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎾᏍᎩᎾᎢ ᎡᎶᎯ.

Yajnavalkya (9th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ ᎤᏓᎷᎸ) ᎠᏒᎾᏍᏗᏱ ᎯᎠ Shatapatha Brahmana, ᎦᏙ ᎤᏍᏗ ᎢᎦᎢ ᎨᏐ geometric ᎢᎦᏛ, ᎠᏠᏯᏍᏗᏍᎩ ᎯᎸᏍᎩ ᎠᏓᏃᎮᏗ π, ᎬᏙᏗ ᎯᎠ closest ᎠᎴᏂᏙᎲ ᎪᏢᎯᏐᏗ 2 ᏎᏍᏗ ᏚᏙᏢᏒᎢ (ᎯᎠ ᎤᎪᏗᏗ ᎧᎵ ᏗᏙᎳᎩ ᏧᎬᏩᎶᏗ π ᎦᎸᎳᏗᏢ Ꮎ ᎢᏳᏩᎪᏗ), ᎠᎴ ᎠᏓᏁᏗ ᎠᏍᏓᏩᏛᏍᏙᏗ ᎪᏣᎸᏙᏗ ᎥᎦᏔᎲᎢ ᎯᎠ ᎣᏩᏒ ᎢᏳᏍᏗ ᎧᏃᎮᏗ.

ᎯᎠ Sulba Sutras ("ᎠᏍᏓᏩᏛᏍᏙᏗ ᎠᏟᎶᏍᏗ" ᎭᏫᎾᏗᏢ Vedic ᎢᏳᏕᏘᏴᏓ), ᎦᏙ ᎤᏍᏗ ᎨᏒᎢ ᏄᏓᎴ ᏚᏙᎥ ᎾᏍᎩᎾᎢ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎨᏒᎩ ᎠᏒᎾᏍᏗᏱ ᎠᏰᎵ 800 ᎤᏓᎷᎸ ᎠᎴ 500 ᎤᏓᎷᎸ ᎠᎴ ᎨᏒᎩ appendices ᎯᎠ Vedas ᎠᎾᏓᏁᎲ ᏗᎫᎪᏔᏅ ᎾᏍᎩᎾᎢ ᎯᎠ ᎠᏁᏍᎨᎲ ᏧᏁᎸᏗᏯ altars. ᎯᎠ Sulba Sutras ᎢᎦᎢ ᎨᏒ ᎯᎠ ᎢᎬᏱ ᎬᏙᏗ ᏎᏍᏗ ᏎᏍᏗ, ᏅᎩ ᎢᏗᎦᏘᎭᎯᎠ ᎤᏙᏢᏒ x² = c ᎠᎴ ᎦᎷᏯᏍᏗ² + bx = c, ᎯᎠ ᎬᏙᏗ ᎯᎠ ᎣᏩᏒ ᎢᏳᏍᏗ ᎧᏃᎮᏗ ᎠᎴ ᏙᎪᏩᎸ ᎣᏩᏒ ᏦᎢ ᎪᎷᏩᏛᏓ algebraically predating Pythagoras, geometric ᎪᎷᏩᏛᏓ ᎦᏌᏆᎸ ᎢᏗᎦᏘᎭ, ᎠᎴ ᏎᏍᏗ geometrical ᏗᎪᎯᏳᏙᏗ. ᎾᏍᎩ ᎯᎠ discoveries ᎠᎴ ᎾᏍᎩ ᎤᎪᏗᏗ ᏄᎵᏍᏔᏅ altar ᎠᏁᏍᎨᎲ, ᎦᏙ ᎤᏍᏗ ᎾᏍᎩ ᎾᏍᏇ ᎤᏗᏅᏒᎩ ᎯᎠ ᎢᎬᏱ ᎤᎾᏅᏛ ᎠᏓᏃᎮᏗ ᎾᏍᎩᎾᎢ ᎯᎠ ᏅᎩ ᏧᏅᏏᏯ ᎤᎾᏍᏕᏢ 2, ᎦᏙ ᎤᏍᏗ ᎨᏒᎩ ᎪᏢᎯᏐᏗ ᎤᏍᏆᏂᎪᏗᏳ 5 ᏎᏍᏗ ᏚᏙᏢᏒᎢ.

Baudhayana (circa 800 ᎤᏓᎷᎸ) ᎠᏒᎾᏍᏗᏱ ᎯᎠ Baudhayana Sulba Sutra, ᎦᏙ ᎤᏍᏗ ᎢᎦᎢ ᎨᏐ ᎧᏃᎮᎸᎯ ᎯᎠ ᎣᏩᏒ ᎢᏳᏍᏗ ᎧᏃᎮᏗ, geometric ᎪᎷᏩᏛᏓ ᎦᏌᏆᎸ ᎢᏗᎦᏘᎭ ᎭᏫᎾᏗᏢ ᏏᏴᏫ ᎤᎧᏁᎳ, ᎯᎸᏍᎩ ᎾᎥᏂᎨ ᎡᎵᏍᏗ π (ᎯᎠ closest ᏧᎬᏩᎶᏗ ᎠᎴᏂᏙᎲ 3.114), ᎨᎳᏛᏍᏗ ᎬᏙᏗ ᎯᎠ ᎢᎬᏱ ᎬᏙᏗ ᏎᏍᏗ ᏗᏎᏍᏗ ᎠᎴ ᏅᎩ ᎠᏓᏃᎮᏗ ᎯᎠ ᏚᏙᏢᏒ ᎦᎷᏯᏍᏗ² = c ᎠᎴ ᎦᎷᏯᏍᏗ² + bx = c, ᎠᎴ ᎠᏓᏃᎮᏗ ᎾᏍᎩᎾᎢ ᎯᎠ ᏅᎩ ᏧᏅᏏᏯ ᎤᎾᏍᏕᏢ 2, ᎦᏙ ᎤᏍᏗ ᏥᏄᏍᏛᎩ ᎪᏢᎯᏐᏗ ᎤᏍᏆᏂᎪᏗᏳ ᎯᏍᎩ ᏎᏍᏗ ᏚᏙᏢᏒᎢ.

Manava (circa 750 ᎤᏓᎷᎸ) ᎠᏒᎾᏍᏗᏱ ᎯᎠ Manava Sulba Sutra, ᎦᏙ ᎤᏍᏗ ᎢᎦᎢ ᎨᏐ ᎾᎥᏂᎨᏍᏙᏗ ᎠᏁᏍᎨᎲ ᎦᏐᏆᎸ ᏂᏛᎴᏅᏓ ᏓᏍᏓᏅᏅ ᏚᎷᏨ, ᎠᎴ ᏅᎩ ᏧᏅᏏᏯ ᏂᏛᎴᏅᏓ ᎦᏐᏆᎸ, ᎦᏙ ᎤᏍᏗ ᎠᏓᏁᏗ ᎾᎥᏂᎨᏍᏙᏗ ᏧᎬᏩᎶᏗᎢ π, ᎬᏙᏗ ᎯᎠ closest ᏧᎬᏩᎶᏗ ᎠᎴᏂᏙᎲ 3.125.

Apastamba (circa 600 ᎤᏓᎷᎸ]) ᎠᏒᎾᏍᏗᏱ ᎯᎠ Apastamba Sulba Sutra, ᎦᏙ ᎤᏍᏗ ᎢᎦᎢ ᎨᏐ ᎯᎠ ᎢᎬᎾᏗ squaring ᎯᎠ ᎦᏐᏆᎸ, ᎠᎦᏎᏍᏙᏗ ᎯᎠ ᎠᎦᏎᏍᏙᏗ ᎢᎬᏁᏗ dividing ᎢᎦᏛ ᎾᎾᎯ 7 ᎢᏗᎦᏗ ᎢᎦᏛᎭ, ᎠᏎᏍᏗᏱ ᎯᎠ ᏅᎩ ᏧᏅᏏᏯ ᎤᎾᏍᏕᏢ 2 ᎪᏢᎯᏐᏗ ᎯᏍᎩ ᏎᏍᏗ ᏚᏙᏢᏒᎢ, ᎤᎦᎾᏭ ᎯᎠ ᏂᎦᎥ ᎦᏌᏆᎸ ᎢᏗᎦᏘᎭ, ᎠᎴ ᎾᏍᎩ ᎾᏍᏇ ᎢᎦᎢ ᎨᏐ ᎠᏓᏃᎮᏗ ᎪᎯᏳᏙᏗ ᎯᎠ ᎣᏩᏒ ᎢᏳᏍᏗ ᎧᏃᎮᏗ, ᎬᏗᏍᎬᎢ ᎡᏍᎦᏂ ᎠᏓᏃᎮᏗ. ᎯᎠ ᎪᏪᎵᏍᎩ Albert Burk ᎠᏆᏤᎵ ᎠᎾᏗᏍᎩ ᎪᎯ ᏥᏄᏍᏛᎩ ᎯᎠ ᎠᎴᏅᏗᏍᎬ ᎪᎯᏳᏙᏗ ᎯᎠ ᎢᏳᏍᏗ ᎧᏃᎮᏗ ᎦᏙ ᎤᏍᏗ Pythagoras copied ᎾᎿ ᎤᏤᎵ ᎠᏓᏩᏛᎯᏓᏍᏗ ᎠᎹ ᎦᏄᎪᎬ.

ᎠᏓᏙᎵᏍᏗ ᎠᎪᎢ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ (ca. 600–300 ᎤᏓᎷᎸ)

ᎾᏍᎩᎾᎢ ᎯᎠ ᎯᎸᎯᏳᎢ ᎠᎪᎢ ᎠᏓᏃᎮᏗ, ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏥᏄᏍᏛᎩ ᎯᎠ ᎠᎵᏍᏚᎶ ᏅᏯᎣᏌᏂ ᎤᎾᏤᎵ agatohvsdi, ᏄᏓᎷᎸᎾ ᎠᏙᎷᏩᏘᏍᎬ ᎤᎾᏤᎵᏛ ᎤᎾᏙᏢᎯ ᎠᏴᏫᏯ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎠᎾᏙᏯᏅᎯᏗᏍᎬ ᏂᎦᏛ ᎠᎴ ᎠᏍᏆᏙᏅᎯ methodology Ꮎ Ꮭ ᏐᎢ ᎤᏩᏂᎦᎸ ᎤᎾᏤᎵ ᎥᎦᏔᎲᎢ ᎠᏰᎲ attained. ᎤᏅᏌ ᎤᏂᎪᏗᏗ ᎯᎠ ᎠᏍᏓᏅᏅ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎤᎪᏗᏗ ᎢᏤ ᏧᏓᎴᏅᏓ ᏗᏎᏍᏗ, ᎠᏓᏲᎲ, ᎦᏚᎢ, ᎠᎴ ᎠᎧᎵᏬᎯ; ᎤᏅᏌ ᎦᏁᏟᏴᏓ Ꮝ methodology ᏂᏛᎴᏅᏓ ᎠᏥᎵᏱᏙᎲ-ᎠᎴ-ᎦᎵᏓᏍᏛ ᎧᏃᎮᎸᏗ ᎪᏣᎴᏛ; ᎤᏅᏌ ᎤᏬᎳᏨ Ꮎ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏗᎦᏎᏍᏙᏗ "ᎢᎪᎯᏛ ᏚᏙᏢᏒ", ᎠᎴ ᎪᏣᎴᏛ, ᎦᏙ ᎤᏍᏗ ᏄᎵᏂᎬᎬ ᏓᎦᏘᎴᎦ ᎠᎴ ᎾᏍᎩ ᎤᏩᏒ ᎾᎥᏂᎨ ᎡᎵᏍᏗ; ᎠᎴ ᎤᏅᏌ ᎤᏙᎷᏩᏛᏓ ᎯᎠ ᎠᏓᏅᏖᏗ "ᎤᏚᎳᏗ ᎪᎷᏩᏛᏗ", ᎦᏙ ᎤᏍᏗ, ᎾᏍᎩᎾᎢ ᎤᏟ ᎢᎦᎢ ᎬᎾᏬᏍᎬ 2000 ᏧᏕᏘᏴᏓ, ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᎠᏰᎸᏅ ᎾᏍᏋ ᎯᎠ ᎤᎵᎶᎯ ᎠᎴᏅᏗᏍᎬ ᎾᏍᎩᎾᎢ ᏂᎦᏛ ᎠᏏᎾᏍᏛ theories.

Thales ᎠᎴ Pythagoras

Pythagorean theorem: a<ᎠᏍᏓᏩᏛᏍᏗ>2</ᎠᏍᏓᏩᏛᏍᏗ> + b<ᎠᏍᏓᏩᏛᏍᏗ>2</ᎠᏍᏓᏩᏛᏍᏗ> = c<ᎠᏍᏓᏩᏛᏍᏗ>2</ᎠᏍᏓᏩᏛᏍᏗ>

Thales (635–543 ᎤᏓᎷᎸ) Miletus (ᎾᏊ ᎭᏫᎾᏗᏢ southwestern ᎬᎾ), ᏥᏄᏍᏛᎩ ᎯᎠ ᎢᎬᏱ ᎦᎪᏃ ᎪᏣᎴᏛ ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎨᏒᎢ ᎠᏠᏯᏍᏔᏅ. ᎾᎿᎢ ᎠᎴ ᎯᏍᎩ geometric ᎠᏍᎪᎳᏗᏍᎬ ᎾᏍᎩᎾᎢ ᎦᏙ ᎤᏍᏗ ᎾᏍᎩ ᎠᏍᎦᏯ ᎤᏬᏪᎳᏅ deductive ᏗᎪᎯᏳᏙᏗ, ᎤᏁᎳᎩ ᎾᏍᎩ ᎾᏍᏊ ᎤᏤᎵ ᏗᎪᎯᏳᏙᏗ ᎤᎭ ᎾᏍᎩ ᏂᎨᏒᎾ survived. Pythagoras (582–496 ᎤᏓᎷᎸ) Ionia, ᎠᎴ ᎣᏂᏯᎨᏍᏙᏗ, Italy, ᎾᎯᏳᎢ colonized ᎾᎥᎢ ᎠᎪᎢ, ᎠᎾᏍᎬᏘ ᎤᎭ ᏭᏪᏙᎢ ᏕᏕᎶᏆᏍᎩ Thales, ᎠᎴ traveled Babylon ᎠᎴ Egypt. ᎯᎠ ᎢᏳᏍᏗ ᎧᏃᎮᏗ Ꮎ ᎯᎸᏍᎩ ᏲᏅ ᎤᏤᎵ ᏚᏙᎥ ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᏂᎨᏒᎾ ᎤᏤᎵ ᎪᎷᏩᏛᏗ, ᎠᎴ ᎾᏍᎩ ᎠᏍᎦᏯ ᏥᏄᏍᏛᎩ ᏄᏓᎷᎸᎾ ᏌᏊ ᎯᎠ ᎢᎬᏱ ᎠᏓᏁᏗ deductive ᎪᎯᏳᏙᏗ ᎾᏍᎩ. ᎾᏍᎩ ᎠᏍᎦᏯ ᎦᏟᏌᏅ ᎤᎾᏓᏟᏌᎲ ᏙᎾᏕ ᎶᏆᏍᎩ ᏴᏩᏚᏫᏛ ᎾᏍᎩ ᎠᏍᎦᏯ ᎠᎦᏎᏍᏙᏗ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᏗᎧᏃᎩᏛ, ᎠᎴ ᎤᏬᎳᏨᎯ, ᎠᎴ ᎢᏧᎳ ᎤᏅᏌ ᎪᎷᏩᏛᏓ ᎤᎪᏗᏗ ᎦᏙ ᎤᏍᏗ ᎦᎸᎳᏗ ᏗᏕᎶᏆᏍᏗ ᏙᎾᏕ ᎶᏆᏍᎩ ᎠᏕᎶᏆᏍᏗ ᎪᎯ ᎢᎦ ᎭᏫᎾᏗᏢ ᎤᎾᏤᎵ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏫᏂᏚᏳᎪᏛ. ᎭᏫᎾᏗᏢ ᎦᏟᏐᏗᎩ, ᎤᏅᏌ ᎪᏢᏅᎯ ᎯᎠ ᏭᏓᎪᎾᏛ ᎪᎷᏩᏛᏗ ᎬᏙᏗ ᏂᎦᏅᎯᏒ ᎠᎴ ᏎᏍᏗ ᏎᏍᏗ.

Plato

Plato (427-347 ᎤᏓᎷᎸ), ᎯᎠ ᏗᏕᎶᏆᏍᎩ ᎤᎪᏗᏗ ᎦᎸᏉᏔᏅ ᎾᎥᎢ ᎯᎠ ᎠᎪᎢ, ᎠᏰᎲ inscribed ᎦᎸᎳᏗᏢ ᎯᎠ ᎠᏴᏍᏙᏗ ᎤᏤᎵ ᏗᎦᏃᏣᎵ ᏗᏕᎶᏆᏍᏗ, "ᎤᎵᏍᎪᎸᏔᏅ ᎧᏂᎩᏛ ᎾᎦᏔᎲᎾ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᏴᏍᏗ ᎠᎭᏂ." ᎤᏁᎳᎩ ᎾᏍᎩ ᎾᏍᏊ ᎾᏍᎩ ᎠᏍᎦᏯ ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᏂᎨᏒᎾ ᎠᏓᏃᎮᏗ ᎤᏩᏒᎠᏍᎦᏯ, ᎤᏤᎵ ᏫᏓᎧᏃᏗᏱ ᎾᎿ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎠᏰᎲ ᎡᏆ ᎾᏓᏛᏂᏌᏁᎲ. ᎠᏓᏃᎮᏗ ᎯᎠ ᎢᏴ accepted ᎤᏤᎵ ᎤᏬᎯᏳᏒ Ꮎ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎢᏳᏗᎾ ᎬᏙᏗ Ꮭ ᎪᎱᏍᏗ ᏗᎬᏔᏂᏓᏍᏗ ᎠᎴ ᎠᎳᏂ ᎠᎦᏓᏗ ᎠᎴ straightedge – ᎥᏝ ᎠᏟᎶᏍᏗ ᎦᏙᎯ ᎪᎱᏍᏗ ᏗᎬᏔᏂᏓᏍᏗ ᏯᏛᎿ ᏥᏄᏍᏗ ᎪᏪᎳᏅᎯ ᏄᎬᏫᏳᏌᏕᎩ ᎠᎴ protractor, ᎢᎬᏂᏏᏍᎩ ᎾᏍᎩ ᎯᎠ ᎨᏒᎩ workman’s ᎪᎱᏍᏗ ᏗᎬᏔᏂᏓᏍᏗ, ᎾᏍᎩ ᏂᎨᏒᎾ ᏧᎬᏩᎶᏗᏯ ᏗᏕᎶᏆᏍᎩ. ᎪᎯ ᎧᏁᎬᎢ ᎤᏗᏅᏒᎩ ᎠᏍᏛᎩ ᎠᎦᏎᏍᏙᏗ ᏰᎵᏊ ᎠᎳᏂ ᎠᎦᏓᏗ ᎠᎴ straightedge ᎠᏁᏍᎨᎲ, ᎠᎴ ᏦᎢ ᎢᏳᏕᏘᏴᏓ ᎠᏁᏍᎨᎲ ᏗᎦᏎᏍᏙᏗ: ᎯᎳᎪ ᎬᏙᏗ ᎾᏍᎩ ᎯᎠ ᎪᎱᏍᏗ ᏗᎬᏔᏂᏓᏍᏗ ᏦᎢᏁ ᏓᏍᏓᏅᏅ ᏚᎷᏨ, ᎠᏛᎯᏍᏙᏗ ᏦᎢ ᏔᎵ ᎢᏳᏩᎫᏗ ᎯᎠ ᎦᏟᏌᏅ ᎪᏪᎵ ᎠᏓᏁᎸ ᏦᎢ, ᎠᎴ ᎠᏛᎯᏍᏙᏗ ᏅᎩ ᏧᏅᏏᏯ ᎢᏗᎦᏗ ᎭᏫᎾᏗᏢ ᎡᏍᎦᏂ ᎠᏓᏁᎸ ᎦᏐᏆᎸ. ᎯᎠ ᏗᎪᎯᏳᏙᏗ ᎯᎠ ᎧᏁᏍᎦ ᎾᏍᎩ ᎯᎠ ᎠᏁᏍᎨᎲ, ᏭᎵᏍᏆᏛ achieved ᎭᏫᎾᏗᏢ ᎯᎠ 19th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ, ᎤᏗᏅᏒᎩ ᎤᎵᏍᎨᏛ ᏚᎵᏍᎪᎵᏴ ᎾᏍᎩ ᎠᏂᏰᎸᏍᎬ ᎯᎠ ᎠᏍᏛᎩ ᎠᏛᎯᏍᏙᏗ ᎯᎠ ᎤᏙᎯᏳ ᎾᏍᎩ ᏎᏍᏗ ᎢᏯᏛᏁᎵᏓᏍᏗ. Aristotle (384-322 ᎤᏓᎷᎸ), Plato’s ᏭᏔᏅ ᏗᏕᎶᏆᏍᎩ, ᎤᏬᏪᎳᏅ ᎧᏃᎮᏍᎩ ᎾᎿ ᎢᏗᎬᎾᏗ ᎤᏂᎦᏛᎲᏍᎩ ᎬᏔᏅᎯ ᎭᏫᎾᏗᏢ deductive ᏗᎪᎯᏳᏙᏗ (ᎠᎪᏩᏛᏗ ᎧᏃᎮᎸᏗ) ᎦᏙ ᎤᏍᏗ ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᏂᎨᏒᎾ ᎠᏏᏴᏫ ᎪᏢᎯᏌᏅ ᎾᏍᎩ ᎢᏳᏩᎪᏗ ᎯᎠ 19th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ.

Hellenistic ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ (ca. 300 ᎤᏓᎷᎸ - 500 ᏂᏓᏙᎳᎬᎾ)

Euclid

ᎠᎨᏴ ᏓᏕᏲᎲᏍᎬ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᏗᏟᎶᏍᏙᏗ ᎾᎾᎢ ᎯᎠ ᎠᏓᎴᏂᏍᎬ ᎢᏳᏕᏘᏴᏓ ᎠᏁᏢᏔᏅᎯ Euclid ᎤᏤᎵ ᎢᏧᏓᎴᎩ, (ca. 1310)

Euclid (ca. 325-265 ᎤᏓᎷᎸ), Alexandria, ᏄᏓᎷᎸᎾ ᏕᏕᎶᏆᏍᎩ ᏌᏊ Plato’s ᏙᎾᏕ ᎶᏆᏍᎩ, ᎤᏬᏪᎳᏅ ᎧᏃᎮᏍᎩ ᎭᏫᎾᏗᏢ 13 ᏗᎪᏪᎵ (ayatolv), ᎢᏴ ᎠᏂᏙᎾᎥ ᎯᎠ ᎢᏧᏓᎴᎩ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎭᏫᎾᏗᏢ ᎦᏙ ᎤᏍᏗ ᎾᏍᎩ ᎠᏍᎦᏯ ᎤᏓᏁᎸᎩ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎭᏫᎾᏗᏢ ᎤᎵᎶᎯ ᎧᏁᎬᎢatic ᎤᏙᏢᏒ, ᎦᏙ ᎤᏍᏗ ᎤᎷᏨᎩ ᎾᏍᏋ ᎤᎾᏅᏛ ᏥᏄᏍᏗ Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎯᎠ ᎧᏃᎮᏍᎩ ᎨᏒᎢ ᎾᏍᎩ ᏂᎨᏒᎾ ᎠᏓᏁᎳᏅᎯ ᏂᎦᏛ Ꮎ ᎯᎠ Hellenistic ᎠᏓᏃᎮᏗ ᎤᎾᏛᎩ ᎾᎾᎢ ᎯᎠ ᎢᏳᏩᎪᏗ ᎬᏩᏚᏫᏛ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ; Euclid ᎤᏩᏒᎠᏍᎦᏯ ᎤᏬᏪᎳᏅ ᏧᏁᎳ ᎤᏟ ᎢᎦᎢ ᎠᏓᏌᎳᏙᏗ ᏗᎪᏪᎵ ᎾᎿ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎢᏧᎳ ᎣᎾᏛ ᏂᏛᎴᏅᏓ ᏐᎢ ᏫᏓᏎᎸᎢ Ꮎ Euclid’s ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᏂᎨᏒᎾ ᎯᎠ ᎢᎬᏱ ᏑᏓᎴᎩ ᏂᏓᏳᏓᎴᏅ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᏓᏓᎶᏙᏗ, ᎠᎴ ᎾᏍᎩ ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᎢᎬᏂᏏᏍᎩ ᎤᏣᏘ ᏄᎬᏫᏳᏌᏕᎩ Ꮎ ᎯᎠ ᎠᏂᏐᎢ ᎤᏅᏨᎩ ᎾᎾᎯ ᎾᏍᏋ ᎠᎴ ᎨᏒᎩ ᎤᎴᎾᎯᏛ. ᎾᏍᎩ ᎠᏍᎦᏯ ᏥᏄᏍᏛᎩ ᎠᏲᎸᎯ ᎯᎠ ᏗᏕᎶᏆᏍᏗ ᎾᎾᎢ Alexandria ᎾᎥᎢ Ptolemy ᎠᏯ, ᎤᎬᏫᏳᎯ Egypt.

ᎯᎠ ᎢᏧᏓᎴᎩ ᎤᎴᏅᎲ ᎬᏙᏗ ᏩᏎᏍᏗ ᎾᎿ ᏓᏓᎴᏂᏍᎬ, ᏄᎬᏫᏳᏒ ᏗᎳᏏᏙᏗ geometric ᏚᎵᏍᎪᎵᏴ (ᎤᏯᏅᎲ ᎧᏁᎬᎢ ᎠᎴ ᎧᏃᎮᎸᏗ), ᎠᎴ ᏂᎦᎥ ᎢᎦᎢ ᏚᎵᏍᎪᎵᏴ (ᎤᏯᏅᎲ ᏧᏣᏔᏊ ᎦᎵᏓᏍᏛ) ᏂᏛᎴᏅᏓ ᎦᏙ ᎤᏍᏗ ᏂᎦᏛ ᎯᎠ ᎠᏣᏪᏐᎸᏍᏙᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏰᎵᏇ ᎾᏍᏋ ᎧᏃᎮᎸᏗ deduced. ᎠᏍᏓᏩᏗᏒ ᎠᎴ ᎤᏤᎵ ᎯᏍᎩ ᎧᏁᎬᎢ, ᎾᏍᎩ ᏯᏛᎾ paraphrased ᎪᏢᏗ ᎯᎠ ᎩᎵᏏ ᎠᎯᏗᎨ ᎥᎪᎵᏰᏍᎬ.

1. ᏂᎦᎵᏍᏗᏍᎬᎫ ᏔᎵ ᏗᎪᏍᏓᏱ ᏰᎵᏇ ᎾᏍᏋ ᎤᎾᏖᎳᏛᎩ ᎾᎥᎢ ᎦᏥᏃᏍᏛ ᎠᏍᏓᏅᏅ.
2. ᏂᎦᎵᏍᏗᏍᎬᎫ ᏎᏍᏗ ᎦᏥᏃᏍᏛ ᎠᏍᏓᏅᏅ ᏰᎵᏇ ᎾᏍᏋ ᎧᏁᏉᏛ ᎭᏫᎾᏗᏢ ᎦᏥᏃᏍᏛ ᎠᏍᏓᏅᏅ.
3. ᎦᏐᏆᎸ ᏰᎵᏇ ᎾᏍᏋ ᎠᏎᏒᏓ ᎬᏙᏗ ᏂᎦᎵᏍᏗᏍᎬᎫ ᎠᏰᎵ ᎠᎴ ᏂᎦᎵᏍᏗᏍᎬᎫ ᎬᎳᏚᏫᏛ.
4. ᏂᎦᏛ ᏚᏳᎪᏛ ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎠᎴ ᎢᏗᎦᏗ ᎠᏂᏏᏴᏫᎭ ᏐᎢ.
5. ᎢᏳᏃ ᏔᎵ ᎦᏥᏃᏍᏛ ᎤᎵᏍᏕᎸᏗ ᎭᏫᎾᏗᏢ ᎦᏃᎯᎵᏙ ᎠᎴ ᏗᎦᎾᏗᏫᏍᏗ ᎾᎥᎢ ᏄᏓᎴ ᎦᏥᏃᏍᏛ ᎠᏍᏓᏅᏅ (ᎤᏯᏅᎲ ᎯᎠ ᎠᎷᏚᎧᏘ), ᎠᎴ ᎯᎠ ᎭᏫᎾᏗᏢ ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎠᏰᎵ ᎯᎠ ᏔᎵ ᎤᎵᏍᏕᎸᏗ ᎠᎴ ᎯᎠ ᎠᎷᏚᎧᏘ ᎠᏝᎥ ᎾᎿ ᏌᏊ ᎠᏍᏆᎨᏂ ᎯᎠ ᎠᎷᏚᎧᏘ ᎦᏟᏐᏗ ᎦᎸᎳᏗᏢ ᎦᏲᎵᎨ ᎬᎾᏬᏍᎬ ᏔᎵ ᏚᏳᎪᏛ ᏓᏍᏓᏅᏅ ᏚᎷᏨ, ᎾᎯᏳᎢ ᎾᎿ Ꮎ ᎠᏍᏆᎨᏂ ᎯᎠ ᎠᎷᏚᎧᏘ, ᎯᎠ ᏔᎵ ᎤᎵᏍᏕᎸᏗ ᎧᏁᏉᏛ ᏫᎵ ᏗᎦᎾᏗᏫᏍᏗ (ᎾᏍᎩ ᎾᏍᏇ ᎤᏯᏅᎲ ᎯᎠ ᏚᏦᏔᏩᏘ ᎧᏃᎮᎸᏗ).

ᎾᏍᎩ ᏥᏄᏍᏛᎩ ᏄᎵᏍᏛ ᎠᎦᏎᏍᏔᏅ, ᎠᎴ Ꮭ ᎤᏝᏏᏛ Euclid ᎤᏩᏒᎠᏍᎦᏯ ᎤᎾᏛᎩ, Ꮎ ᎤᏤᎵ ᎯᏍᎩᏁ ᎧᏁᎬᎢ ᏰᎵᏇ ᎾᏍᏋ ᏅᎪᏢᎯᏌᏅ ᎾᎥᎢ ᎯᎠ ᏍᏆᎵᎯᎨᏍᏗ ᎧᏃᎮᎸᎯ “Given ᎠᏍᏓᏅᏅ ᎠᎴ ᎪᏍᏓᏱ ᎾᏍᎩ ᏂᎨᏒᎾ ᎾᎿ ᎯᎠ ᎠᏍᏓᏅᏅ, ᎾᎿᎢ ᎨᏒᎢ ᎾᏍᎩ ᎤᏩᏒ ᏌᏊ ᎠᏍᏓᏅᏅ ᏗᎬᏩᎶᏒ ᎯᎠ ᎠᏓᏁᎸ ᎪᏍᏓᏱ ᎠᎴ ᎭᏫᎾᏗᏢ ᎯᎠ ᎤᏠᏱ ᎦᏃᎯᎵᏙ ᎬᏙᏗ ᎯᎠ ᎠᏓᏁᎸ ᎠᏍᏓᏅᏅ Ꮎ ᎾᏛᏁ ᎾᏍᎩ ᏂᎨᏒᎾ ᏗᎦᎾᏗᏫᏍᏗ ᎯᎠ ᎠᏓᏁᎸ ᎠᏍᏓᏅᏅ.” ᎪᎯ ᎨᏒᎢ ᎤᏯᏅᎲ Playfair’s ᎧᏁᎬᎢ, ᎤᎶᏐᏅ ᎯᎠ ᎩᎵᏏ ᏗᏕᏲᎲᏍᎩ ᎦᎪ ᎧᏁᎢᏍᏔᏅ ᎪᏢᏗ ᎯᎠ ᏅᎪᏢᎯᏐᏗᏱ ᎭᏫᎾᏗᏢ ᏂᎦᏛ ᎯᎠ ᏗᏕᎶᏆᏍᏗ ᎠᏓᏓᎶᏙᏗ.

ᎯᎠ ᎧᏁᎬᎢ, ᎧᎵ ᏗᏙᎳᎩ Plato, ᎢᏳᏗᎾ ᎾᏍᏋ ᏄᏦᏍᏛᎾ ᎠᎴ ᎣᏩᏒ-ᎥᏝ ᏳᏓᎷᎳ ᏚᎵᏍᎪᎵᏴ, ᎾᏍᎩ ᎢᎬᏂᏏᏍᎩ ᏰᎵ ᏧᎸᏌᏓ ᎤᏙᎯᏳ Ꮎ ᎤᏅᏌ ᎤᏚᎳᏗ Ꮭ ᎪᎯᏳᏙᏗ. Euclid’s ᎢᎬᏱ ᏅᎩ ᎧᏁᎬᎢ ᏗᏠᎯᏍᏗ ᎪᎯ ᎠᎦᏛᏗ ᏄᏍᏛ, ᎠᎴ ᎯᎠ ᎯᏍᎩᏁ, ᎢᏧᎳᎭ ᎢᏳᏃ ᏅᎪᏢᎯᏌᏅ ᎾᎥᎢ Playfair’s ᎧᏁᎬᎢ, ᎨᏒᎢ ᎾᏍᎩ ᏂᎨᏒᎾ ᏄᏦᏍᏛᎾ, ᎠᎴ ᎤᎪᏗᏗ ᏯᏓᎢᏗᏏ ᎯᏁᎩ ᎾᏍᎩ ᏂᎨᏒᎾ ᎣᏩᏒ-ᎥᏝ ᏳᏓᎷᎳ ᎾᏍᎩᏯᎢ ᎯᎠ ᎢᎬᏱ ᏅᎩ. ᎯᎠ ᎯᏍᎩᏁ resembled ᎤᏟ ᎢᎦᎢ ᎯᎠ ᎢᏳᏍᏗ ᎧᏃᎮᏗ Ꮎ Euclid ᎪᎯᏳᏔᏅ ᏂᏛᎴᏅᏓ ᎯᎠ ᎧᏁᎬᎢ. ᎤᏗᏗᏢᎢᎨᏍᏙᏗ, Euclid ᎤᏙᎷᏩᏛᏓ ᎬᏂᎨᏒ ᏄᏍᏛᎢ ᎢᎦᏛ ᎤᏤᎵ ᎪᎷᏩᏛᏗ ᏦᎢ ᏄᏠᏯᏍᏛᎾ ᎬᏗᏍᎬᎢ ᎯᎠ ᎯᏍᎩᏁ ᎧᏁᎬᎢ. ᎯᎠ ᎤᏝᏅᏓᏕᎲ ᏚᎴᏅ, ᏄᏓᎷᎸᎾ ᎾᎯᏳᎨᏒ Euclid’s ᎠᎴᏫᏍᏙᏗ, Ꮎ ᎯᎠ ᎯᏍᎩᏁ ᎧᏁᎬᎢ ᏰᎵᏇ ᎠᎴ ᎢᏳᏗᎾ ᎾᏍᏋ ᎪᎯᏳᏔᏅ ᏥᏄᏍᏗ ᎢᏳᏍᏗ ᎧᏃᎮᏗ ᏂᏛᎴᏅᏓ ᎯᎠ ᎢᎬᏱ ᏅᎩ, ᎠᎴ ᎯᎠ ᎢᏴ ᎨᏒᎢ ᎡᎵᏉ ᏥᏄᏍᏗ ᎧᏁᎬᎢ. ᎯᎠ ᎢᏴ ᎤᎴᏅᎲ ᎤᎪᏗᏗ centuries ᎠᏁᎸᏙᏗ ᎪᎯᏳᏙᏗ ᎯᎠ ᎯᏍᎩᏁ ᎧᏁᎬᎢ, ᎠᎴ ᎯᎠ ᎠᏛᏛᎲᏍᎩ ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᏂᎨᏒᎾ ᏗᏓᏙᎳᏤᏗ ᎢᏳᏩᎪᏗ ᎯᎠ 19th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ.

Archimedes

Archimedes (287-212 ᎤᏓᎷᎸ), Syracuse, Sicily, ᎯᎳᎪ ᎢᏳ ᎾᏍᎩ ᏥᏄᏍᏛᎩ ᎠᎪᎢ ᎦᏚᎲᎢ-ᎤᏔᏂᏗ ᎦᏙᎯ, ᎨᏒᎢ ᎢᏳᏓᎵᎭ ᎠᎦᏎᏍᏔᏅ ᎾᏍᏋ ᎯᎠ ᏭᏔᏅ ᎯᎠ ᎠᎪᎢ ᎠᏓᏃᎮᏗ, ᎠᎴ ᎢᏳᏓᎵᎭᏊ ᎢᏧᎳᎭ ᏓᎪᎥᎩ ᏥᏄᏍᏗ ᏌᏊ ᎯᎠ ᏦᎢ ᏭᏔᏅ ᏂᎦᏛ ᎢᏳᏩᎪᏗ (ᎨᎳᏛᏍᏗ ᎬᏙᏗ Isaac Newton ᎠᎴ Carl Friedrich Gauss). ᎠᏰᎲ ᎾᏍᎩ ᎠᏍᎦᏯ ᎾᏍᎩ ᏂᎨᏒᎾ ᏭᏪᏙᎢ ᎠᏓᏃᎮᏗ, ᎾᏍᎩ ᎠᏍᎦᏯ ᏯᏓᎢᏗᏏ ᏙᎢ ᎾᏍᏋ ᎠᏅᏓᏛᎯ ᏥᏄᏍᏗ ᎡᏆ ᎢᏳᏍᏗ ᏯᏛᎿ, ᎠᏥᎸ ᏗᎯᎴᎩ, ᎠᎴ ᎪᏪᎵ ᎠᏪᎵ ᎪᏪᎵᏍᏗ. ᎭᏫᎾᏗᏢ ᎤᏤᎵ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᎾᏍᎩ ᎠᏍᎦᏯ ᎤᏙᎷᏩᏛᏓ ᎢᏗᎬᎾᏗ ᎤᏙᎯᏳ ᎤᏠᏱ ᎯᎠ ᎢᏗᎦᏘᎭ ᎢᏯᏛᏁᎵᏓᏍᏗ ᎠᏓᏃᎮᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎠᎴ ᎯᎠ ᎠᏎᎸᎯ ᏧᎵᎬᏩᎳᏅᎯ ᏂᎦᏛ ᎠᏓᏃᎮᏗ. ᎯᎠ ᎾᏍᎩ ᎤᏩᏒ ᏑᏓᎴᎩ ᎦᎷᎶᎩ ᎾᏍᎩᎾᎢ ᎯᎠ ᏗᏁᏝᎾ ᎾᏍᎩ ᎯᎠ ᏕᎦᎶᎨᏒ ᏥᏄᏍᏛᎩ ᎠᏏᎾᏍᏛ ᎠᏓᏃᎮᏗ ᎠᏅᏓᏗᏍᏙᏗ ᎪᏪᎶᏗ ᎭᏫᎾᏗᏢ ᎦᏙ ᎤᏍᏗ ᎢᎦᏪᏍᏗ ᎤᏤᎵ ᎠᏓᏅᏖᏗ.

Archimedes ᎠᏰᎲ ᎠᏍᏓᏩᏛᏓ Eudoxian ᎢᏗᎬᎾᏗ ᎪᏪᎶᏗ ᎠᏥᏄᏉᏫᏍᎬ geometric ᎪᎷᏩᏛᏓ. ᏌᏊ ᎪᎷᏩᏛᏓ ᎯᎠ ᎡᏍᎦᏂ ᎠᎴ ᎦᏟᏌᏅ ᎪᏪᎵ ᎠᏗᏌᏓᏗᏍᏗ ᎬᏔᏅᎯ ᏌᏊᎭ ᎠᏍᏓᏩᏛᏍᏗ, ᎤᏙᏢᏒ ᏚᏳᎪᏛ ᏗᏎᏍᏗ ᎠᏅᏓᏗᏍᏙᏗ ᎪᏪᎶᏗ Ꮎ ᏥᏄᏍᏛᎩ ᎪᏢᏅᎯ ᎾᎥᎢ Egyptians 1,700 ᏧᏕᏘᏴᏓ ᎢᎬᏱᎨᏍᏙᏗ. ᏌᏊᎭ ᎠᏍᏓᏩᏛᏍᏗ ᏕᎬᏔᏛ ᎠᏰᎵ Archimedes' ᎢᎬᎾᏗ slicing ᎯᎠ ᎠᏗᏌᏓᏗᏍᏗ ᎾᎾᎯ ᎤᏍᏗ ᏗᎬᎭᎷᏴᎯ, creating ᎯᎠ ᎢᎬᏱ ᎤᏙᏢᏒ ᎠᏓᏃᎮᏗ, ᏥᏄᏍᏗ ᎠᏓᏁᎸ ᎾᎥᎢ ᎯᎠ ᎪᎯᏳᏙᏗ (ᎪᏪᎶᏗ ᎾᎥᎢ Dijksterhuis)

4A/3 = + /3 + /12

ᎠᎴ, Ꮝ 1/4th geometric ᎢᎪᎯᏓ ᎨᏒ ᎧᏁᏉᎥᎢ ᎤᏠᏱᎭ ᎤᏙᏢᏒ

4A/3 = + /4 + /16 + /64 + ... /(4n) + ...

ᎯᎠ Moscow ᏚᏳᎪᏛ ᎠᏃᏪᎵᏍᎬ, ᎠᏁᏎᎯᎲ ᎢᎪᎯ 2,000 ᎤᏓᎷᎸ ᎾᏍᎩ ᎾᏍᏇ sliced ᎯᎠ ᎡᏍᎦᏂ ᎠᏟᎶᏍᏗ ᎤᏓᏂᏝᏅ, ᎧᎵ ᏗᏙᎳᎩ ᎥᏩᏘᏍᎬ Ꮝ ᎡᏍᎦᏂ, ᏥᏄᏍᏗ Archimedes ᎣᏂᏯᎨᏍᏙᏗ ᎢᎬᎾᏔᏅᎯ ᎾᎥᎢ ᎠᏍᏓᏩᏗᏒ ᎯᎠ Eudoxian 1/4th geometric ᎧᏁᏉᎥᎢ ᎤᏠᏱᎭ, ᎠᎴ proving ᎤᏤᎵ ᏄᎵᏍᏔᏅ ᎾᎥᎢ ᏌᏊᎭ ᎠᏍᏓᏩᏛᏍᏗ ᏗᏎᏍᏗ.

ᎤᎶᏐᏅ Archimedes

ᎤᎶᏐᏅ Archimedes, Hellenistic ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎤᎴᏅᎲ ᎠᏓᏯᏍᏗ. ᎾᎿᎢ ᎨᏒᎩ ᎦᏲᎵ ᎠᏲᎵ ᎠᏂᏃᏈᏏ ᎩᎳ ᎦᎷᏨ, ᎠᎴ ᎯᎠ ᏓᎶᏂᎨᏍᏙᏗ ᎢᏳᏕᏘᏴᏓ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏥᏄᏍᏛᎩ ᎦᏬᎯᎸᏙᏗ. Proclus (410-485), ᎪᏪᎵ ᎠᏪᎵ ᎪᏪᎵᏍᏗ ᏩᏎᏍᏗ ᎾᎿ ᎾᎿ ᎯᎠ ᎢᎬᏱ ᎪᏪᎵ Euclid, ᏥᏄᏍᏛᎩ ᏌᏊ ᎯᎠ ᎣᏂ ᎤᎵᏍᎨᏛ ᏗᎾᏁᎶᎲᏍᎩ ᎭᏫᎾᏗᏢ Hellenistic ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎾᏍᎩ ᎠᏍᎦᏯ ᏥᏄᏍᏛᎩ ᏰᎵ ᎢᏯᏛᏁᎯ ᎠᏓᏃᎮᏗ, ᎠᎴ ᎤᏟ ᎢᎦᎢ ᎤᎵᏍᎨᏛ, ᎾᏍᎩ ᎠᏍᎦᏯ ᏥᏄᏍᏛᎩ ᎤᏬᏚ ᏗᎫᎪᏔᏅ ᎾᎿ ᎯᎠ ᏧᏂᎸᏫᏍᏓᏁᏗ Ꮎ preceded ᎾᏍᎩ ᎠᏍᎦᏯ. ᎤᏣᏘ Ꮎ ᏗᎦᎸᏫᏍᏓᏁᏗ ᏄᏛᏁᎸ ᎾᏍᎩ ᏂᎨᏒᎾ ᏩᎵᏃᎯᏯᏍᎬ ᎪᎯᏊ ᎢᏴ ᏥᎩ ᎢᏧᎳᎪᏗ, ᎠᎴ ᎨᏒᎢ ᎤᎾᏅᏛ ᎢᏧᎳ ᎠᏴ ᎾᏍᎩ ᎤᏩᏒ ᏗᎬᏩᎶᏒ ᎤᏤᎵ ᏩᏎᏍᏗ ᎾᎿ. ᎯᎠ ᎶᎻ ᎠᏰᎵ ᎤᏙᏢ ᎠᎴ ᏓᎳᏏᏛ Ꮎ succeeded ᎠᎴ ᎠᎦᏎᏍᏙᏗ ᎯᎠ ᎠᎪᎢ ᎦᏚᎲᎢ-ᎧᏃᎮᎭ ᎠᏛᎯᏍᏔᏅ ᎤᎵᎶᎲᏍᎩ ᎠᏥᎸ ᏗᎯᎴᎩ, ᎠᎴ Ꮭ ᎠᏓᏃᎮᏗ ᎠᏓᏚᎬ ᎪᏪᎵ.

ᎢᎬᏱ 'agatohvsdi,' ᎾᏍᎩᎾ ᎨᏒ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᎴ ᎡᎶᎯ/ᏗᎦᏃᏣᎳᏅ, ᏥᏄᏍᏛᎩ ᏗᎵᎪᏔᏅ ᎯᎠ ᏩᎦᎸᎳᏗᏴ ᎾᏍᎩᎾᎢ ᎤᎪᏗᏗ ᎢᏳᏕᏘᏴᏓ ᏗᏕᎶᏆᏍᎩ. ᎯᎠ ᎠᎳᏂ ᎠᎦᏓᏗ ᎭᏫᎾᏗᏢ ᎪᎯ 13th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ ᎠᏃᏪᎵᏍᎬ ᎨᏒᎢ ᏗᎬᏟᎶᏍᏙᏗ ᎤᏁᎳᏅᎯ ᎤᏤᎵ ᎢᏯᏛᏁᎯ ᏗᏁᏝᎾ, ᏥᏄᏍᏗ ᎤᎪᏗᏗ ᎤᏬᎯᏳᏅ Ꮎ ᎾᎿᎢ ᏥᏄᏍᏛᎩ ᎪᎱᏍᏗ ᎢᎦᏛ "ᏩᎦᎸᎳᏗᏴ" ᎠᎴ "ᎧᎵᏬᎯ" Ꮎ ᏰᎵᏇ ᎾᏍᏋ ᎠᏩᏛᏗ ᎭᏫᎾᏗᏢ ᎦᏐᏆᎸ

ᎯᎠ ᎡᏆ ᎪᏪᎵ ᏕᎦᏅ ᏗᎪᎵᏰᏗ Alexandria ᏥᏄᏍᏛᎩ ᎣᏂᏯᎨᏍᏙᏗ ᎤᎴᏴᏒ. ᎾᎿᎢ ᎨᏒᎢ ᎠᏛᏏᏒ ᎪᎯᏳᏙᏗ ᏄᎾᏛᏅ ᎪᏪᎵᏍᎩ Ꮎ ᎯᎠ ᎪᏪᎵ ᏕᎦᏅ ᏗᎪᎵᏰᏗ Alexandria ᏄᏓᎷᎸᎾ ᎠᎩᎵᏲᏨ ᏂᏛᎴᏅᏓ ᎯᎸᏍᎩ ᎤᏲᏨᎯ ᏂᏕᎦᎵᏍᏔᏂᏙᎲ, ᎠᎴ Ꮎ ᎯᎠ ᎤᏲᏨᎯ Alexandria ᎤᏤᎵ ᎦᎵᏓᏍᏛ ᏧᏂᎳᏫᎢᏍᏗ ᎭᏫᎾᏗᏢ ᎯᎠ ᎣᏂᏱᏳ 4th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ ᏥᏄᏍᏛᎩ ᏄᏓᎷᎸᎾ ᎯᎠ ᎤᎪᏗᏗ ᎤᎶᏒᏍᏔᏅᎯ ᎠᎴ ᏭᎵᏍᏆᏙᏅ ᏌᏊ. ᎯᎠ ᎪᎯᏳᏙ ᎾᏍᎩᎾᎢ Ꮎ ᎤᏲᏨᎯ ᎨᏒᎢ ᎯᎠ ᎤᎪᏗᏗ ᎤᎵᏍᏛ ᎠᎴ ᏗᎦᏂᏴᏗ. ᎡᎯᏍᏛ ᎤᏤᎵ ᎠᏴᏍᏙᏗ ᎠᎾᏍᎬᏘ ᎠᏔᎴᏒ ᎠᎹᏱ ᎤᎭ ᎤᏗᏅᏒᎩ ᎯᎠ ᎤᏲᎱᏎᎸ ᎢᎦᏛ 40,000-70,000 ᎪᏪᎸ ᎭᏫᎾᏗᏢ ᎪᎱᏍᏗ ᎠᏗ ᎤᏓᏃᏢᎢ ᎯᎠ ᏥᏳ ᏗᏔᎳᏗᏍᏗ (ᏥᏄᏍᏗ Luciano Canfora ᎠᏘᏲᎯᎭ, ᎤᏅᏌ ᎨᏒᎩ ᏄᏓᎷᎸᎾ ᎯᎸᏍᎩ ᏗᏟᎶᏍᏔᏅ ᎠᏛᎯᏍᏔᏅ ᎾᎥᎢ ᎯᎠ ᎪᏪᎵ ᏕᎦᏅ ᏗᎪᎵᏰᏗ ᏄᏪᎵᏒ ᎾᏍᎩᎾᎢ ᏩᏗ ᏙᏱᏗᏢ), ᎠᎴ ᎾᏍᎩ ᎨᏒᎢ ᎧᏁᏍᎦ ᎤᎭ ᎦᎸᏉᏔᏅ ᎯᎠ ᎪᏪᎵ ᏕᎦᏅ ᏗᎪᎵᏰᏗ ᎠᎴ ᎤᏓᏂᏝᏅ, ᎠᏓᏁᎸ Ꮎ ᎾᎿᎢ ᎨᏒᎢ ᏰᎵᏊ ᎢᎦᎢ ᎪᎯᏳᏙ Ꮎ ᎢᏧᎳ existed ᎣᏂᏯᎨᏍᏙᏗ.

ᎤᎵᎶᎯ ᎢᎬᏁᎯ ᏓᏄᏩ, ᎠᎦᏲᎳᏗᏍᏗ ᎫᏢᎥᏍᎬ ᎭᏫᎾᏗᏢ ᎠᏍᏆᏂᎪᏙᏗ ᎠᎴ ᎠᏯᏍᏗ ᎠᎩᏍᏗ ᎢᏤ ᎪᏪᎸ ᎠᎴ ᏂᎦᎥᏊ ᎢᏳᏕᏘᏴᏓ ᎤᏁᏉᏨ ᎭᏫᎾᏗᏢ ᎬᏙᏗ-ᏧᏁᎸᏗᏯ ᎤᎾᏄᎪᎢᏍᏗ ᏄᏓᎷᎸᎾ contributed ᎠᎦᏲᎳᏗᏍᏗ ᎭᏫᎾᏗᏢ ᎯᎠ ᎠᏰᎸ ᏭᏚᎸᏛ ᎬᏙᏗ ᏰᎵ ᎠᏩᏛᏗ ᎭᏫᎾᏗᏢ ᎯᎠ ᎪᏪᎵ ᏕᎦᏅ ᏗᎪᎵᏰᏗ, ᎾᏍᎩ ᎨᏒᎢ ᎭᏫᎾᏗᏢ ᎯᎠ ᏅᎩᏁ ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ. ᎯᎠ Serapeum ᏥᏄᏍᏛᎩ ᎤᏙᎯᏳᎯᏯ ᎠᏲᏍᏔᏅ ᎾᎥᎢ Theophilus ᎭᏫᎾᏗᏢ 391, ᎠᎴ ᎯᎠ ᎤᏓᏂᏝᏅ ᎠᎴ ᎪᏪᎵ ᏕᎦᏅ ᏗᎪᎵᏰᏗ ᎠᎾᏍᎬᏘ ᎤᎭ ᎤᎶᏒᎢ ᎠᏥᏐᏅᏅ ᎯᎠ ᎤᏠᏱ togiyadv dunilvwisdanehv.

ᎢᏍᎳᎻᎩ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ (ca. 700 - 1500)

ᎤᏆᏓᏛ ᏂᏛᎴᏅᏓ ᎯᎠ al-Jabr Wa-l-Muqabilah

ᎯᎠ Islamic ᏄᎬᏫᏳᏌᏕᎩᎤᎬᎩ (ᎢᏍᎳᎻᎩ ᏓᎳᏏᏛ) ᎪᏢᏅ ᏗᎦᎾᏗᎯᏍᏗ ᎯᎠ ᎠᏰᎵ ᎧᎸᎬ, ᏧᏴᏢ ᎬᎿᎦᏍᏛ, Spain, Portugal, ᎠᏫᎨᏂᏍᏖᏂ ᎠᎴ ᎢᎦᏛᎭ Pakistan, ᎤᎴᏅᎲ ᏴᏩᏚᏫᏛ 640 CE. ᎢᏍᎳᎻᎩ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎾᎯᏳᎨᏒ ᎪᎯ ᎠᎴᏫᏍᏙᏗ ᏥᏄᏍᏛᎩ ᎢᎬᏱᏗᏢ ᎠᏓᏃᎮᏗ ᎤᏟ ᎬᏰᎸᏗ ᎬᎾᏬᏍᎬ geometric, ᎤᏁᎳᎩ ᎾᏍᎩ ᎾᏍᏊ ᎾᎿᎢ ᎨᏒᎩ ᎤᎵᏍᎨᏛ ᏧᏂᎸᏫᏍᏓᏁᏗ ᎾᎿ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᏛᏕᎶᏆᏍᎬ ᎭᏫᎾᏗᏢ ᏳᎳᏛ ᎠᏓᏱᎸ ᎠᎴ ᏭᎵᏱᎶᎸ ᎯᎠ Hellenistic ᏧᏂᎸᏫᏍᏓᏁᏗ ᏧᏩᏓᏔᏅᏒ ᎨᏒᎩ ᎤᎴᎾᎯᏛ ᎠᏂ, ᎠᎴ survived ᎾᏍᎩ ᎤᏩᏒ ᎭᏫᎾᏗᏢ ᎯᎠ ᎢᏍᎳᎻᎩ ᎠᏰᎵ ᎠᎾᏕᎶᏆᏍᎬ.

ᎾᏍᎩ ᎤᏁᎳᎩ ᎯᎠ Muslim ᎠᏓᏃᎮᏗ ᎠᎴ ᎤᎪᏗᏗ ᏂᎬᎢ ᎾᏍᎩᎾᎢ ᎤᎾᏤᎵ ᏗᎦᎸᏫᏍᏓᏁᏗ ᎾᎿ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᏎᏍᏗ ᎪᎷᏩᏛᏗ ᎠᎴ ᏎᏍᏗ ᎢᏯᏛᏁᎵᏓᏍᏗ, ᎤᏅᏌ ᎾᏍᎩ ᎾᏍᏇ ᎪᏢᏅᎯ ᎤᎪᏗ ᏗᎵᏍᎪᎸᏙᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎠᎴ ᏚᏳᎪᏛ ᎡᎶᎯ, ᎠᎴ ᎨᏒᎩ ᎤᎾᏚᏓᎸ ᎾᏍᎩᎾᎢ ᎯᎠ ᏚᏙᎳᏩᏛᎲ ᎠᏓᏃᎮᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. Geometrical ᎢᎦᎢ ᎨᏒᎩ ᎠᏓᏁᎸᎯ ᏥᏄᏍᏗ "ᎠᏓᏃᎮᏗ ᏓᎦᏘᎴᎦ" ᎾᎥᎢ ᎤᎪᏗᏗ Muslim ᎠᏓᏃᎮᏗ ᏱᏂᎬᏛᎾ.

ᎯᎠ ᎦᏂᏓᏛ Muḥammad ibn Mūsā al-Ḵwārizmī (ᎤᏕᏅ 780) undertook ᎠᏓᏅᏍᏗ ᎠᏔᏲᏍᏙᏗ ᏗᏎᏍᏗ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᏗᏎᏍᏗ, ᎢᏧᎳ ᏓᏍᏓᏅᏅ ᏚᎷᏨ, ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎯᎠ Euclidean ᎪᎷᏩᏛᏗ ᏗᏎᏍᏗ, ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎠᎴ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏗᏎᏍᏗ ᎤᎬᏩᎵ. ᎪᎯ ᏥᏄᏍᏛᎩ ᎯᎳᎪ ᎯᎠ ᏗᏁᏝᎾ polynomial ᏗᏎᏍᏗ ᎤᎬᏩᎵ, combinatorial ᏗᎫᎪᏙᏗ ᎢᎬᏁᏗ, ᎠᏓᏃᎮᏗ ᏗᎫᎪᏙᏗ ᎢᎬᏁᏗ, ᎯᎠ ᎠᏓᏃᎮᏗ ᎪᎷᏩᏛᏓ ᎠᏓᏃᎮᏗ, ᎯᎠ ᎢᏤ ᏑᏓᎴᎩ ᏂᏓᏳᏓᎴᏅ ᎪᎷᏩᏛᏗ ᏗᏎᏍᏗ, ᎠᎴ ᎯᎠ geometric ᎠᏁᏍᎨᎲ ᎠᏓᏃᎮᏗ ᏚᎴᏅ.

Al-Mahani (ᎤᏕᏅ 820) ᏁᎭ ᎯᎠ ᎠᏓᏅᏖᏗ ᎠᎦᏲᎳᏗᏍᎩ geometrical ᏗᎦᏎᏍᏙᏗ ᏯᏛᎿ ᏥᏄᏍᏗ duplicating ᎯᎠ ᏦᎢ ᏗᎦᏎᏍᏙᏗ ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ. al-Karaji (ᎤᏕᏅ 953) ᎠᎧᎵᏬᎯ ᎠᏎᏊᎢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᏂᏛᎴᏅᏓ geometrical ᎠᏥᏰᎶᎲ ᎠᎴ ᏅᎪᏢᎯᏌᏅ ᎠᏂ ᎬᏙᏗ ᎯᎠ ᏗᏎᏍᏗ ᎦᎾᏄᎪᏫᏒ ᏗᎦᎪᏗ ᎠᏥᏰᎶᎲ ᎦᏙ ᎤᏍᏗ ᎠᎴ ᎾᎾᎢ ᎯᎠ ᎠᏰᎵ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎪᎯ ᎢᎦ.

Thabit ibn Qurra

ᎾᏍᎩ ᎤᏁᎳᎩ Thabit ibn Qurra (ᎤᏕᏅ 836) contributed ᏎᏍᏗ ᎡᏍᎦᏂ ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᎭᏢᏃ ᎾᏍᎩ ᎠᏍᎦᏯ ᏚᎾᏁᎶᏅ ᎤᎵᏍᎨᏛ ᎠᏛᏁᎵᏍᎩ ᎤᏍᏆᎸᎡᎲ ᎭᏫᎾᏗᏢ ᎠᏛᏅᎢᏍᏗᏍᎩ ᎯᎠ ᎦᎶᎯᏍᏗ ᎾᏍᎩᎾᎢ ᏯᏛᎿ ᎤᎵᏍᎨᏛ ᏚᏳᎪᏛ discoveries ᏥᏄᏍᏗ ᎯᎠ ᎧᏁᏉᎥᎢ ᎯᎠ ᎠᏓᏅᏖᏗ ᏎᏍᏗ (ᎤᏙᎯᏳ) ᎤᏙᎯᏳ ᎾᏍᎩ ᏎᏍᏗ, ᏂᎦᏛ ᎠᏓᏃᎮᏗ, ᎢᏳᏍᏗ ᎧᏃᎮᏗ ᎭᏫᎾᏗᏢ ᎦᏐᏆᎸ ᏓᏍᏓᏅᏅ ᏚᎷᏨ, ᎠᏓᏃᎮᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎠᎴ ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎭᏫᎾᏗᏢ ᎡᎶᎯ Thabit ᏥᏄᏍᏛᎩ ᏌᏊ ᎯᎠ ᎢᎬᏱ ᏗᎾᏓᏅᏘᏐᏗᏍᎩ ᎯᎠ Ptolemaic ᎢᏯᏛᏁᎵᏓᏍᏗ, ᎠᎴ ᎭᏫᎾᏗᏢ ᎠᏥᏅᏏᏓᏍᏗ ᎾᏍᎩ ᎠᏍᎦᏯ ᏥᏄᏍᏛᎩ ᎠᏩᏥᎩ ᏄᎵᏂᎬᎬ.

ᎤᎵᏍᎨᏛ geometrical ᎤᏔᏂᏗ ᎦᏙᎯ Thabit ᎤᏤᎵ ᏗᎦᎸᏫᏍᏓᏁᏗ ᏥᏄᏍᏛᎩ ᎤᏤᎵ ᎪᏪᎵ ᎾᎿ ᎯᎠ ᎪᏪᎳᏅ ᎠᎪᎵᏰᏗ ᎠᎾᎵᏐᏈᎸᏍᎬ. ᎭᏫᎾᏗᏢ ᎪᎯ ᎪᏪᎵ, Thabit ᏗᏓᏅᏓᏁᏗᏱ ᎬᏙᏗ ᎠᏓᏃᎮᏗ ᎠᏥᏰᎶᎲ ᎢᎬᎾᏔᏅᎯ ᎠᎾᎵᏐᏈᎸᏍᎬ geometrical ᎤᎪᏗᏗ. ᎯᎠ ᎠᎪᎢ ᎠᏰᎲ dealt ᎬᏙᏗ geometric ᎤᎪᏗᏗ ᎠᎴ ᎠᏰᎲ ᎾᏍᎩ ᏂᎨᏒᎾ ᏁᎵᏒ ᎠᏂ ᎭᏫᎾᏗᏢ ᎯᎠ ᎤᏠᏱ ᎦᎶᎯᏍᏗ ᏥᏄᏍᏗ ᏗᏎᏍᏗ ᎦᏙ ᎤᏍᏗ ᎯᎠ ᎤᏠᏱ ᏗᎫᎪᏔᏅ ᏗᏎᏍᏗ ᏰᎵᏇ ᎾᏍᏋ ᎢᎬᎾᏔᏅᎯ. ᎾᎥᎢ introducing ᎠᏓᏃᎮᏗ ᎠᏥᏰᎶᎲ ᎾᎿ ᎤᎪᏗᏗ ᏧᏩᎫᏔᏅᏒ ᎾᏍᎩ ᎠᏰᎸᏅ ᏥᏄᏍᏗ geometric ᎠᎴ ᎬᏙᏗ-ᎠᏓᏃᎮᏗ, Thabit ᎤᏂᎩᏍᏔᏅ ᏫᏚᏳᎪᏛ ᎦᏙ ᎤᏍᏗ ᎤᏗᏅᏒᎩ ᏭᎵᏱᎶᎸ ᎯᎠ generalisation ᎯᎠ ᏎᏍᏗ ᎠᏓᏅᏖᏗ.

ᎭᏫᎾᏗᏢ ᎢᎦᏛ ᎪᎯᏳᎲᏍᎦ, Thabit ᎨᏒᎢ ᎤᏓᏚᎯᏌᏘ ᎯᎠ ᎠᏓᏅᏖᏍᎬ Plato ᎠᎴ Aristotle, ᎾᏍᎩᎾ ᎨᏒ ᎾᏍᎩ ᎠᏂᏰᎸᏍᎬ ᎠᏌᎳᏙᏗ. ᎾᏍᎩ ᏯᏓᎢᏗᏏ ᎢᎦᏰᎵᏍᏗ Ꮎ ᎠᎭᏂ ᎤᏤᎵ ᎠᏓᏅᏖᏍᎬ ᎠᎴ ᏚᎳᏏᏔᏅᎩ ᎾᎿ ᎪᎯᏳᏙᏗ ᎬᏗᏍᎬᎢ ᏓᏃᏏᏏᏍᎬ ᎾᏍᎩ ᎠᏂᏰᎸᏍᎬ ᎠᏌᎳᏙᏗ ᎭᏫᎾᏗᏢ ᎤᏤᎵ geometrical ᏓᏃᏏᏏᏍᎬ.

ᎤᎶᏐᏅ Thabit ibn Qurra

Ibrahim ibn Sinan (ᎤᏕᏅ 908), ᎦᎪ ᎬᏂᎨᏒ ᏄᏩᏁᎸ ᎢᎬᎾᏗ ᏂᎦᏛ ᎤᏟ ᎢᎦᎢ ᏂᎦᎥ ᎬᎾᏬᏍᎬ Ꮎ Archimedes, ᎠᎴ al-Quhi (ᎤᏕᏅ 940) ᎨᏒᎩ ᏓᏓᏘᏂᏒ ᏗᏎᏍᏗ ᎭᏫᎾᏗᏢ ᎠᎾᎴᏂᏍᎬᎢ ᎠᎴ ᎦᏟᏐᏗᎩ ᎠᎪᎢ ᎦᎸᎳᏗᎨᏍᏙᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎭᏫᎾᏗᏢ ᎯᎠ ᎢᏍᎳᎻᎩ ᎡᎶᎯ. ᎾᏍᎩ ᎯᎠ ᎠᏓᏃᎮᏗ, ᎠᎴ ᎭᏫᎾᏗᏢ ᎾᏍᎩᎾ al-Haytham, ᎤᎦᏎᏍᏔᏅᎩ ᎠᏨᏍᏛ ᎠᎴ investigated ᎯᎠ ᎠᎪᏩᏛᏗ properties ᏓᏎᏘ ᎪᏢᏅᎯ ᏂᏛᎴᏅᏓ ᏒᏙᏂ ᎾᎿ ᎨᏒ.

ᎡᎶᎯ, ᎢᏳᏩᎪᏗ-ᎤᏍᏆᏂᎪᏛ ᎠᎴ ᎠᏓᏕᏲᎲᏍᎩ ᎡᎶᎯ ᎠᏓᏁᎳᏅ ᏐᎢ motivations ᎾᏍᎩᎾᎢ geometrical ᎠᎴ trigonometrical ᏅᎠᏯᏍᏗ. ᎾᏍᎩᎾᎢ ᏱᏓᏟᎶᏍᏔᏅ Ibrahim ibn Sinan ᎠᎴ ᎤᏤᎵ ᎤᏚᏚ Thabit ibn Qurra ᎢᏧᎳ ᎤᎦᏎᏍᏔᏅᎩ ᎠᏓᏲᎲ ᎧᏁᏨ ᎢᏯᏛᏁᏗ ᎭᏫᎾᏗᏢ ᎯᎠ ᎠᏁᏍᎨᎲ ᎢᏳ ᎢᎪᎯ. Abu'l-Wafa ᎠᎴ Abu Nasr Mansur ᎢᏧᎳ ᎢᎬᎾᏔᏅᎯ ᎦᏐᏆᎸ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎡᎶᎯ.

Omar Khayyám

Omar Khayyám (ᎤᏕᏅ 1048) ᏥᏄᏍᏛᎩ Persian ᎠᏓᏃᎮᏗ, ᏥᏄᏍᏗ ᎠᏔᎴᏒ ᎠᎹᏱ ᏥᏄᏍᏗ ᏗᎪᏪᎵᏍᎩ ᏗᎧᏃᎮᎸᏍᎩ. ᎨᎳᏛᏍᏗ ᎬᏙᏗ ᎤᏤᎵ ᏗᎦᏃᏣᎵ ᏥᏄᏍᏗ ᏗᎪᏪᎵᏍᎩ ᏗᎧᏃᎮᎸᏍᎩ, ᎾᏍᎩ ᎠᏍᎦᏯ ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᎾᏍᏇ ᏗᎦᏃᏣᎵ ᎾᎯᏳᎨᏒ ᎤᏤᎵ ᎠᎴᏫᏍᏙᏗ ᏥᏄᏍᏗ ᎠᏓᏃᎮᏗ, ᎠᏔᎴᏒ ᎠᎹᏱ ᎤᎾᏅᏛ ᎾᏍᎩᎾᎢ ᎪᎷᏩᏛᏗ ᎯᎠ ᏂᎦᎥ ᎢᎬᎾᏗ solving cubic ᎢᏗᎦᏘᎭᎾᎥᎢ ᏗᎦᎾᏗᏫᏍᏗ ᎠᏗᏌᏓᏗᏍᏗ ᎬᏙᏗ ᎦᏐᏆᎸ. ᎭᏫᎾᏗᏢ ᎦᏟᏐᏗᎩ ᎾᏍᎩ ᎠᏍᎦᏯ ᎪᎷᏩᏛᏓ ᎯᎠ ᏔᎵ ᎤᏚᏃᎯᏍᏔᏍᎧ, ᎠᎴ authored ᎠᏙᏚᎯᏍᏙᏗ Euclid ᎤᏤᎵ theories ᏚᏦᏔᏩᏘ ᎦᏙ ᎤᏍᏗ ᎪᏢᏅᎯ ᎤᎾᏤᎵ ᎦᎶᎯᏍᏗ England, ᎭᏢᏃ ᎤᏅᏌ contributed ᎯᎠ ᎤᏩᎫᏗᏗᏒ ᏚᏙᎳᏩᏛᎲ ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. Omar Khayyam ᎾᏍᎩ ᎾᏍᏇ ᎦᏟᏌᏅ ᎯᎠ ᎬᏙᏗ ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎠᎴ ᎾᎥᏂᎨ ᎡᎵᏍᏗ ᎪᎷᏩᏛᏗ ᎠᏓᏁᎳᏁᏗ ᎢᏗᎬᎾᏗ solving ᎠᏓᏃᎮᏗ ᎠᏓᏃᎮᏗ ᎾᎥᎢ geometrical ᎤᏅᏔᏂᏓᏍᏗ. ᎾᏍᎩ ᎠᏍᎦᏯ ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᎤᎪᏗᏗ ᎤᎾᏚᏓᎸ ᎾᏍᎩᎾᎢ ᎯᎠ ᏚᏙᎳᏩᏛᎲ ᎠᏓᏃᎮᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ.

ᎭᏫᎾᏗᏢ Ꭺ ᎪᏪᎳᏅᎯ ᎾᎥᎢ Khayyam ᎤᏓᎷᎸ ᎤᏤᎵ ᏗᎦᏃᏣᎵ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᏓᎵᏍᏛ ᎧᏃᎮᏍᎩ ᎾᎿ ᎬᏂᎨᏒ ᎾᏅᏁᎲ ᏗᎦᏎᏍᏙᏗ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᎾᏍᎩ ᎠᏍᎦᏯ ᎠᎦᏎᏍᏙᏗ ᎯᎠ ᎠᎦᏎᏍᏙᏗ ᎢᎬᏁᏗ: ᎠᏩᏛᏗ ᎪᏍᏓᏱ ᎾᎿ ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎦᏐᏆᎸ ᎭᏫᎾᏗᏢ ᏯᏛᎿ ᏄᏍᏛᎢ Ꮎ ᎯᎳᎪ ᎢᏳ ᏄᎶᏒᏍᏛᎾ ᎨᏒᎢ ᎤᏬᎭᏒᎩ ᏂᏛᎴᏅᏓ ᎯᎠ ᎪᏍᏓᏱ ᏌᏊ ᎯᎠ ᏩᎦᏛ radii, ᎯᎠ ᎠᎾᎵᏐᏈᎸᏍᎬ ᎯᎠ ᏄᎶᏒᏍᏛᎾ ᎤᏤᎵ ᏂᎦᏅᎯᏒ Ꮎ ᎯᎠ ᎬᎳᏚᏫᏛ ᎢᏗᎦᏗ ᎯᎠ ᎠᎾᎵᏐᏈᎸᏍᎬ ᎯᎠ ᎢᎦᏛ ᏕᎫᎪᏔᏅ ᎾᎥᎢ ᎯᎠ ᎤᎳᏏᏕᎾ ᎯᎠ ᏄᎶᏒᏍᏛᎾ. Khayyam ᏓᎾᏛᏁᎵᏍᎬ Ꮎ ᎪᎯ ᎠᎦᏎᏍᏙᏗ ᎢᎬᏁᏗ ᎨᏒᎢ ᎢᏗᎦᏘᎭ solving ᏔᎵᏁ ᎠᎦᏎᏍᏙᏗ ᎢᎬᏁᏗ: ᎠᏩᏛᏗ ᏚᏳᎪᏛ ᏦᎢ ᎤᎲᎢ ᎯᎠ ᏧᎬᏩᎶᏗ ᏂᎬᎿᏅ Ꮎ ᎯᎠ hypotenuse ᎢᏗᎦᏗ ᎯᎠ ᎢᎦᎢ ᏌᏊ ᎦᏅᏍᎨᏂ ᎧᏁᏉᏨ ᎯᎠ ᏂᎦᏛᎢ ᎾᎿ ᎯᎠ hypotenuse. ᎪᎯ ᎠᎦᏎᏍᏙᏗ ᎢᎬᏁᏗ ᎭᏫᎾᏗᏢ ᎠᎦᏔᎲᏍᏗ ᎤᏗᏅᏒᎩ Khayyam ᎤᎦᎾᏭ ᎯᎠ cubic ᎢᏗᎦᏘᎭ x³ + 200x = 20x² + 2000 ᎠᎴ ᎾᏍᎩ ᎠᏍᎦᏯ ᎠᏩᏛᏗ ᎤᏙᎯᏳ ᎤᎾᏍᏕᏢ ᎪᎯ cubic ᎾᎥᎢ ᎠᎦᏎᏍᏗᏍᎬ ᎯᎠ ᏗᎦᎾᏗᏫᏍᏗ ᏄᎶᏒᏍᏛᎾ ᎠᏗᏌᏓᏗᏍᏗ ᎠᎴ ᎦᏐᏆᎸ. ᎾᎥᏂᎨᏍᏙᏗ ᎠᏓᏃᎮᏗ ᎪᎷᏩᏛᏓ ᏥᏄᏍᏛᎩ ᎾᎯᏳᎢ ᎠᏩᏛᏗ ᎾᎥᎢ ᏧᏠᎯᏍᏗ ᎭᏫᎾᏗᏢ trigonometric ᏗᎦᏍᎩᎶ. ᎠᏎᏱᎩ ᎢᏧᎳᎭ ᎤᏟ ᎢᎦᎢ ᎤᏍᏆᏂᎪᏗᏳ ᎨᏒᎢ ᎯᎠ ᏯᏛᎾ ᎤᏙᎯᏳ Ꮎ Khayyam ᎧᏃᎮᎭ Ꮎ ᎯᎠ ᎪᎷᏩᏛᏓ ᎪᎯ cubic ᎧᏁᎦ ᎢᏯᏛᏁᏗ ᎯᎠ ᎬᏙᏗ ᏒᏙᏂ ᎾᎿ ᏕᎨᏒ ᎠᎴ Ꮎ ᎾᏍᎩ ᏝᏰᎵ ᎾᏍᏋ ᎤᏍᏆᏂᎪᏗ ᎾᎥᎢ ᎠᎳᏂ ᎠᎦᏓᏗ ᎠᎴ straightedge, ᏄᎵᏍᏔᏅ ᎦᏙ ᎤᏍᏗ ᏯᏓᎢᏗᏏ ᎾᏍᎩ ᏂᎨᏒᎾ ᎾᏍᏋ ᎪᎯᏳᏔᏅ ᎾᏍᎩᎾᎢ ᏄᏓᎴ 750 ᏧᏕᏘᏴᏓ.

ᎢᏯᏓᏛᏁᎸ ᎾᎥᎢ Albrecht Dürer featuring Mashallah, ᏂᏛᎴᏅᏓ ᎯᎠ ᎦᏓ ᎦᏂᏴᏙ ᎤᏆᏓᏛ ᎯᎠ ᏧᎬᏩᎶᏗ ᎠᎦᏘᏯ scientia motus orbis (Latin ᏅᎬᎪᏔᏅᎯ ᎬᏙᏗ ᎢᏯᏓᏛᏁᎸ, 1504). ᏥᏄᏍᏗ ᎭᏫᎾᏗᏢ ᎤᎪᏗᏗ ᎢᏳᏕᏘᏴᏓ ᏗᏟᎶᏍᏙᏗ, ᎯᎠ ᎠᎳᏂ ᎠᎦᏓᏗ ᎠᎭᏂ ᎨᏒᎢ ᎢᏯᏓᏛᏁᎸ ᏗᏁᎸᏙᏗ ᏥᏄᏍᏗ ᎠᏔᎴᏒ ᎠᎹᏱ ᏥᏄᏍᏗ ᎠᎦᏙᎲᏍᏗ, ᎭᏫᎾᏗᏢ ᏩᏎᎸᎯ ᎤᏁᎳᏅᎯ ᏥᏄᏍᏗ ᎯᎠ ᎪᏪᎵ ᎠᏪᎵ ᎪᏪᎵᏍᏗ ᏗᏁᏝᎾ

ᎤᏤᎵ ᎧᏃᎮᏍᎩ ᎾᎿ ᎬᏂᎨᏒ ᎾᏅᏁᎲ ᏗᎦᏎᏍᏙᏗ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎢᎦᎢ ᎨᏒᎩ ᎧᎵᏬᎯ ᏗᏓᎴᎾᏔᏅ ᏗᎦᏘᏌᏅᎩ cubic ᎠᏓᏃᎮᏗ ᎬᏙᏗ geometric ᎪᎷᏩᏛᏓ ᎠᏩᏛᏗ ᎾᎥᎢ ᎤᏅᏔᏂᏓᏍᏗ ᏗᎦᎾᏗᏫᏍᏗ ᏒᏙᏂ ᎾᎿ ᏕᎨᏒ. ᎭᏫᎾᏗᏢ ᏯᏛᎾ ᎤᏙᎯᏳ Khayyam ᎠᏓᏁᏗ ᎤᏍᏆᏂᎪᏗ ᎧᏃᎮᏍᎩ ᏄᎵᏍᏔᏅ ᎬᏂᎨᏒ ᎢᎬᏁᏗ ᎭᏫᎾᏗᏢ ᎦᏙ ᎤᏍᏗ ᎾᏍᎩ ᎠᏍᎦᏯ ᎠᏆᏤᎵ ᎠᎾᏗᏍᎩ Ꮎ ᎯᎠ ᎠᎪᎢ ᎠᏰᎲ ᎠᎦᏍᎦᏂ ᎥᏝ ᎪᎱᏍᏗ ᎾᎿ ᎯᎠ ᎪᎷᏩᏛᏗ cubic ᎠᏓᏃᎮᏗ. ᎤᏙᎯᏳᎢ, ᏥᏄᏍᏗ Khayyam ᎪᏪᎵᎠ, ᎯᎠ ᏗᎵᏍᎪᎸᏙᏗ ᎾᎥᎢ ᎢᎬᏱᎨᏍᏙᏗ ᎠᏃᏪᎵᏍᎩ ᏯᏛᎿ ᏥᏄᏍᏗ al-Mahani ᎠᎴ al-Khazin ᎨᏒᎩ ᎠᏁᏢᏙᏗ geometric ᏗᎦᏎᏍᏙᏗ ᎾᎾᎯ ᎠᏓᏃᎮᏗ ᎠᏓᏃᎮᏗ (ᎪᎱᏍᏗ ᎦᏙ ᎤᏍᏗ ᏥᏄᏍᏛᎩ ᎠᏏᏴᏫ ᏰᎵ ᏂᎨᏒᎾ ᎤᏓᎷᎸ ᎯᎠ ᏗᎦᎸᏫᏍᏓᏁᏗ Muḥammad ibn Mūsā al-Ḵwārizmī). ᏱᏂᎬᏛᎾ, Khayyam ᎤᏩᏒᎠᏍᎦᏯ ᎤᏁᎵᏒᎯ ᎤᎭ ᏭᏪᏙᎢ ᎯᎠ ᎢᎬᏱ ᎤᎷᏅᏗ ᏂᎦᎥ ᎪᎷᏩᏛᏗ cubic ᎠᏓᏃᎮᏗ.

ᎭᏫᎾᏗᏢ Commentaries ᎾᎿ ᎯᎠ ᎤᏦᏍᏗ ᎧᏃᎮᎸᏗ Euclid ᎤᏤᎵ ᎪᏪᎵ Khayyam ᎪᏢᏅᎯ ᎠᎾᏓᏁᎲ ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎾᏍᎩ ᎤᏁᎳᎩ ᎪᎯ ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᏂᎨᏒᎾ ᎤᏤᎵ ᎢᏰᎵᏍᏗ. ᎭᏫᎾᏗᏢ ᎠᏁᎶᏗᏍᎬ ᎪᎯᏳᏙᏗ ᎯᎠ ᏚᏦᏔᏩᏘ ᎧᏃᎮᎸᏗ ᎾᏍᎩ ᎠᏍᎦᏯ ᎤᏝᏅᏓᏕᎲ ᎪᎯᏳᏔᏅ properties ᏗᏎᏍᏗ ᎭᏫᎾᏗᏢ ᎬᏙᏗ-Euclidean geometries. Khayyam ᎾᏍᎩ ᎾᏍᏇ ᎤᏓᏁᎸ ᎤᎵᏍᎨᏛ ᏂᏚᎵᏍᏔᏅ ᎾᎿ ᎠᎾᎵᏐᏈᎸᏍᎬ ᎭᏫᎾᏗᏢ ᎪᎯ ᎪᏪᎵ, ᏫᎧᏁᏉᎬ Euclid ᎤᏤᎵ ᏗᎦᎸᏫᏍᏓᏁᏗ ᎠᏠᏯᏍᏛ ᎯᎠ ᎠᏓᏃᎮᏗ ᎠᎾᎵᏐᏈᎸᏍᎬ. ᎯᎠ ᏭᎵᏍᎨᏗᏴ Khayyam ᎤᏤᎵ ᎠᎾᏓᏁᎲ ᎨᏒᎢ Ꮎ ᎾᏍᎩ ᎠᏍᎦᏯ ᎠᎦᏎᏍᏔᏅ ᎢᏧᎳ Euclid ᎤᏤᎵ ᏩᏎᏍᏗ ᎾᎿ ᎢᏗᎦᏘᎭ ᎠᎾᎵᏐᏈᎸᏍᎬ (ᎦᏙ ᎤᏍᏗ ᏥᏄᏍᏛᎩ Ꮎ ᎢᎬᏱ ᎧᏁᎢᏍᏔᏅ ᎾᎥᎢ Eudoxus) ᎠᎴ ᎯᎠ ᏩᏎᏍᏗ ᎾᎿ ᎢᏗᎦᏘᎭ ᎠᎾᎵᏐᏈᎸᏍᎬ ᏥᏄᏍᏗ ᎧᏁᎢᏍᏔᏅ ᎾᎥᎢ ᎢᎬᏱᎨᏍᏙᏗ ᎢᏍᎳᎻᎩ ᎠᏓᏃᎮᏗ ᏯᏛᎿ ᏥᏄᏍᏗ al-Mahani ᎦᏙ ᎤᏍᏗ ᏥᏄᏍᏛᎩ ᏚᎳᏏᏔᏅᎩ ᎾᎿ ᎤᎵᏱᎸᏛ ᎠᏍᏓᏩᏛᏍᏗ. Khayyam ᎪᎯᏳᏔᏅ Ꮎ ᎯᎠ ᏔᎵ ᏩᏎᏍᏗ ᎾᎿ ᎠᎴ ᎢᏗᎦᏘᎭ. ᎾᏍᎩ ᎠᏍᎦᏯ ᎾᏍᎩ ᎾᏍᏇ posed ᎯᎠ ᎠᏛᏛᎲᏍᎩ ᎢᏳᏃ ᎠᎾᎵᏐᏈᎸᏍᎬ ᏰᎵᏇ ᎾᏍᏋ ᎾᏍᎩ ᎠᏰᎸᏅ ᏥᏄᏍᏗ ᏎᏍᏗ ᎠᎴ ᏧᎦᎶᎦ ᎯᎠ ᎠᏛᏛᎲᏍᎩ ᎬᏂᎨᏒ ᎾᏅᏁᎲ.

Sharafeddin Tusi

Persian ᎠᏓᏃᎮᏗ Sharafeddin Tusi (ᎤᏕᏅ 1135) ᏄᏛᏁᎸ ᎾᏍᎩ ᏂᎨᏒᎾ ᎠᏍᏓᏩᏛᏍᏗ ᎯᎠ ᏂᎦᎥ ᏚᏙᎳᏩᏛᎲ Ꮎ ᎤᎷᏨᎩ ᏗᎬᏩᎶᏒ al-Karaji ᎤᏤᎵ ᏗᏕᎶᏆᏍᏗ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎠᎴ ᎤᏟ ᎬᏰᎸᏗ ᎠᏍᏓᏩᏛᏓ Khayyam ᎤᏤᎵ ᎠᏔᏲᏍᏙᏗ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎾᏍᎩ ᎠᏍᎦᏯ ᎤᏬᏪᎳᏅ ᎧᏃᎮᏍᎩ ᎾᎿ cubic ᎠᏓᏃᎮᏗ, ᎦᏙ ᎤᏍᏗ ᎾᏓᏛᏁ ᎤᎵᏍᎨᏛ ᎠᎾᏓᏁᎲ ᏄᏓᎴ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎦᏙ ᎤᏍᏗ aimed ᎠᎦᏎᏍᏙᏗ ᎠᏓᏲᎲ ᎾᎥᎢ ᎤᏅᏔᏂᏓᏍᏗ ᎠᏓᏃᎮᏗ, ᎯᎠ ᎢᏴ inaugurating ᎯᎠ ᎠᎦᏎᏍᏙᏗ ᎠᏓᏃᎮᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ.

17th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ

ᎯᎳᎪ ᎢᏳ ᏳᎳᏛ ᎤᎴᏅᎲ ᏗᏓᏲᎯᏍᏗ ᏂᏛᎴᏅᏓ Ꮝ ᎤᎳᏏᎩ ᎤᎧᏁᎳ, ᎯᎠ Hellenistic ᎠᎴ ᎢᏍᎳᎻᎩ ᏓᎵᏍᏛ ᎾᎿ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᏩᏛᏗ ᎭᏫᎾᏗᏢ ᎢᏍᎳᎻᎩ libraries ᎨᏒᎩ translated ᏂᏛᎴᏅᏓ Arabic ᎾᎾᎯ Latin. ᎯᎠ ᏚᏳᎪᏛ deductive ᎢᏗᎬᎾᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᏩᏛᏗ ᎭᏫᎾᏗᏢ Euclid’s ᎢᏧᏓᎴᎩ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎨᏒᎩ relearned, ᎠᎴ ᎤᏗᏗᏢ ᏚᏙᎳᏩᏛᎲ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎭᏫᎾᏗᏢ ᎯᎠ ᏧᎾᏣᏅᏙᏗ ᎢᏧᎳ Euclid (Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ) ᎠᎴ Khayyam (ᎠᏓᏃᎮᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ) ᎤᎵᏱᎸᏛ, ᏄᎵᏍᏔᏅ ᎭᏫᎾᏗᏢ ᎬᏩᎶᏒᏍᏗ ᎢᏤ ᎢᏳᏍᏗ ᎧᏃᎮᏗ ᎠᎴ ᎠᏓᏅᏖᏗ, ᎤᎪᏗᏗ ᎠᏂ ᎤᏙᎯᏳ ᏭᏓᎪᎾᏛ ᎠᎴ ᎤᎵᎶᎯ.

ᏓᏕᏲᎲᏍᎬ ᎾᎿ ᎢᎬᎾᏗ ᎾᎥᎢ Rene Descartes

ᎭᏫᎾᏗᏢ ᎯᎠ ᎢᎬᏱ 17th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ, ᎾᎿᎢ ᎨᏒᎩ ᏔᎵ ᎤᎵᏍᎨᏛ ᏚᎾᏙᎷᏩᏛᎲ ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎯᎠ ᎢᎬᏱ ᎠᎴ ᎤᎪᏗᏗ ᎤᎵᏍᎨᏛ ᏥᏄᏍᏛᎩ ᎯᎠ ᏗᏁᏝᎾ ᎠᏓᏃᎮᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎠᎴ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎬᏙᏗ ᎠᏟᎶᏍᏗ ᎦᏙᎯ ᎠᎴ ᎠᏓᏃᎮᏗ, ᎾᎥᎢ Rene Descartes (1596-1650) ᎠᎴ Pierre ᏧᎬᏩᎶᏗ ᎠᎦᏘᏯ Fermat (1601-1665). ᎪᎯ ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᎢᏳᎵᏍᏙᏗ ᏭᎵᏍᎨᏗᏴ ᎯᎠ ᏚᏙᎳᏩᏛᎲ ᎠᏓᏃᎮᏗ ᎠᎴ ᏚᏳᎪᏛ ᎢᎦᎢ ᎠᎦᏙᎲᏍᏗ ᎢᏳᏍᏗ ᏯᏛᎿ. ᎯᎠ ᏔᎵᏁ geometric ᏚᏙᎳᏩᏛᎲ ᎪᎯ ᎠᎴᏫᏍᏙᏗ ᏥᏄᏍᏛᎩ ᎯᎠ ᎠᏓᏅᏍᏗ ᎠᎦᏎᏍᏙᏗ projective ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎾᎥᎢ Girard Desargues (1591-1661). Projective ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎨᏒᎢ ᎯᎠ ᎠᎦᏎᏍᏙᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏄᏠᏯᏍᏛᎾ ᎠᏓᏃᎮᏗ, ᎣᏍᏛ ᎯᎠ ᎠᎦᏎᏍᏙᏗ ᎯᎳᎪ ᏗᎪᏍᏓᏱ align ᎬᏙᏗ ᎠᏂᏏᏴᏫᎭ ᏐᎢ. ᎾᎿᎢ ᎠᏰᎲ ᏭᏪᏙᎢ ᎢᎦᏛ ᎢᎬᏱ ᏗᎦᎸᏫᏍᏓᏁᏗ ᎭᏫᎾᏗᏢ ᎪᎯ ᎡᏍᎦᏂ ᎾᎥᎢ Hellenistic ᎠᏓᏃᎮᏗ, ᏂᎬᎢ Pappus (ca. 340). ᎯᎠ ᏭᏔᏅ ᎠᏛᎯᏍᏙᏗ ᎯᎠ ᏠᎨᏏ ᏄᎵᏍᏔᏅᎩ ᎬᏙᏗ Jean-ᎦᏣᏄᎳ Poncelet (1788-1867).

ᎭᏫᎾᏗᏢ ᎯᎠ ᎣᏂᏱᏳ 17th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ, ᎠᏓᏃᎮᏗ ᏥᏄᏍᏛᎩ ᎤᏙᎷᏩᏛᏓ ᎤᎾᏤᎵᏛ ᎤᎾᏙᏢᎯ ᎠᎴ ᎾᎥᏂᎨᏍᏗ ᎠᎵᎪᏁᏗ ᎾᎥᎢ Isaac Newton (1642-1727) ᎠᎴ Gottfried Wilhelm von Leibniz (1646-1716). ᎪᎯ ᏥᏄᏍᏛᎩ ᎯᎠ ᎠᏓᎴᏂᏍᎬ ᎢᏤ ᏠᎨᏏ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎾᏊ ᎤᏯᏅᎲ ᏗᎫᎪᏙᏗ ᎢᎬᏁᏗ. ᎤᏁᎳᎩ ᎾᏍᎩ ᎾᏍᏊ ᎾᏍᎩ ᏂᎨᏒᎾ ᎤᏩᏌᏊ ᎤᏩᏂᎦᎸ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎾᏍᎩ ᎨᏒᎢ ᎠᎾᎵᏐᏈᎸᏍᎬ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎠᎴ ᎾᏍᎩ ᎤᏍᏆᏂᎪᏗ ᏔᎵ ᎯᎸᏍᎩ ᎢᏯᏓᏁᎸ ᏗᎦᏎᏍᏙᏗ Ꮎ ᎠᏰᎲ ᎦᏅᎯᏓ ᏭᏪᏙᎢ ᎾᎥᏂᎨᏍᏗ ᎠᎵᏁᎩ ᎪᏪᎵ: ᎥᏩᏘᏍᎬ ᏚᏓᏂᏴᏒ ᎤᎵᏍᏕᎸᏗ ᏧᏓᎴᎿᎢ ᎠᏓᏲᎲ, ᎠᎴ ᎥᏩᏘᏍᎬ ᎡᏍᎦᏂ ᎠᏠᏯᏍᏔᏅ ᎾᎥᎢ ᎾᏍᎩ ᎠᏓᏲᎲ. ᎯᎠ ᎢᏗᎬᎾᏗ ᎠᏓᏃᎮᏗ ᎠᎦᏲᎳᏛᎯ ᎾᏍᎩ ᎯᎠ ᏗᎦᏎᏍᏙᏗ ᎾᏍᎩ ᎤᎪᏗᏗ ᎢᏚᏳᎪᏛ ᎢᏳᏍᏗ ᏱᏓᏛᎿ ᎠᏓᏃᎮᏗ.

ᎯᎠ 18th ᎠᎴ 19th centuries

ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ

ᎯᎠ ᎠᎦᏴᎵ ᎠᎦᏎᏍᏙᏗ ᎢᎬᏁᏗ proving Euclid’s ᎯᏍᎩᏁ ᎧᏃᎮᎸᏗ, ᎯᎠ "ᏚᏦᏔᏩᏘ ᎧᏃᎮᎸᏗ", ᏂᏛᎴᏅᏓ ᎤᏤᎵ ᎢᎬᏱ ᏅᎩ ᎧᏃᎮᎸᏗ ᎠᏰᎲ ᎥᏝ ᏭᏪᏙᎢ ᎬᎨᏫᏒᎯ. ᎠᏓᎴᏂᏍᎬ ᎾᏍᎩ ᏂᎨᏒᎾ ᎦᏅᎯᏓ ᎤᎶᏐᏅ Euclid, ᎤᎪᏗᏗ ᎠᏁᎳᏛᏅ ᎬᏂᎨᏒ ᎾᏅᏁᎲ ᎨᏒᎩ ᎠᏓᏁᎸ, ᎠᎴ ᏂᎦᏛ ᎨᏒᎩ ᎣᏂᏯᎨᏍᏙᏗ ᎠᏩᏛᏗ ᎾᏍᏋ ᎦᏬᏂᎯᏍᏗ ᎠᏕᎶᏆᏍᏗ, ᏗᎬᏩᎶᏒ ᎠᎵᏍᎪᎸᏙᏗ ᎾᎾᎯ ᎯᎠ ᎤᏂᎦᏛᎲᏍᎩ ᎢᎦᏛ ᎤᎵᏍᎪᎵᏴ ᎦᏙ ᎤᏍᏗ ᎤᏩᏌᏊ ᎠᏰᎲ ᎾᏍᎩ ᏂᎨᏒᎾ ᏭᏪᏙᎢ ᎪᎯᏳᏔᏅ ᏂᏛᎴᏅᏓ ᎯᎠ ᎢᎬᏱ ᏅᎩ ᎧᏃᎮᎸᏗ. ᎤᏁᎳᎩ ᎾᏍᎩ ᎾᏍᏊ Omar Khayyám ᏥᏄᏍᏛᎩ ᎾᏍᎩ ᎾᏍᏇ ᎤᏄᎸᎲᏍᎩ ᎭᏫᎾᏗᏢ proving ᎯᎠ ᏚᏦᏔᏩᏘ ᎧᏃᎮᎸᏗ, ᎤᏤᎵ ᎠᏙᏚᎯᏍᏙᏗ Euclid ᎤᏤᎵ theories ᏚᏦᏔᏩᏘ ᎠᎴ ᎤᏤᎵ ᎪᎯᏳᏙᏗ properties ᏗᏎᏍᏗ ᎭᏫᎾᏗᏢ ᎬᏙᏗ-Euclidean geometries contributed ᎯᎠ ᎤᏩᎫᏗᏗᏒ ᏚᏙᎳᏩᏛᎲ ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎾᎥᎢ 1700 ᎡᏆ ᏗᏓᏅᏓᏁᏗᏱ ᎠᏰᎲ ᏭᏪᏙᎢ ᎪᎷᏩᏛᏓ ᎬᏩᏚᏫᏛ ᎦᏙ ᎤᏍᏗ ᏰᎵᏇ ᎾᏍᏋ ᎪᎯᏳᏔᏅ ᏂᏛᎴᏅᏓ ᎯᎠ ᎢᎬᏱ ᏅᎩ, ᎠᎴ ᎦᏙ ᎤᏍᏗ ᎯᎠ ᎤᏍᏆᏂᎪᏗ ᎨᏒᎩ ᎭᏫᎾᏗᏢ ᎠᏁᎶᏗᏍᎩ ᎪᎯᏳᏙᏗ ᎯᎠ ᎯᏍᎩᏁ. Saccheri, Lambert, ᎠᎴ Legendre ᎠᏂᏏᏴᏫᎭ ᏄᏛᏁᎸ ᎤᎵᎶᎲᏍᎩ ᏗᎦᎸᏫᏍᏓᏁᏗ ᎾᎿ ᎯᎠ ᎠᎦᏎᏍᏙᏗ ᎢᎬᏁᏗ ᎭᏫᎾᏗᏢ ᎯᎠ 18th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ, ᎠᎴ ᏙᎢ ᎤᏅᏨᎩ ᏍᏆᎳᎢ ᎦᏣᏄᎳ. ᎭᏫᎾᏗᏢ ᎯᎠ ᎢᎬᏱ 19th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ, Gauss, Johann Bolyai, ᎠᎴ Lobatchewsky, ᎠᏂᏏᏴᏫᎭ ᎤᎾᏤᎵᏛ ᎤᎾᏙᏢᎯ, ᎤᎩᏒᎩ ᏄᏓᎴᎿᎥ ᎤᎾᏄᎪᎢᏍᏗ. ᎠᏓᎴᏂᏍᎬ ᎠᏓᎳᏩᏎᏗ Ꮎ ᎾᏍᎩ ᏥᏄᏍᏛᎩ ᏰᎵ ᏂᎨᏒᎾ ᎪᎯᏳᏙᏗ ᎯᎠ ᏚᏦᏔᏩᏘ ᎧᏃᎮᎸᏗ, ᎤᏅᏌ ᎠᏫᏒᏗ ᎠᏥᏄᏉᏫᏍᎬ ᎤᏙᎷᏬᏗ ᎣᏩᏒ-ᎤᏠᏱᎭ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎭᏫᎾᏗᏢ ᎦᏙ ᎤᏍᏗ Ꮎ ᎧᏃᎮᎸᏗ ᏥᏄᏍᏛᎩ ᎦᎶᏄᎮᏛ. ᎭᏫᎾᏗᏢ ᎪᎯ ᎤᏅᏌ ᎨᏒᎩ ᎦᏣᏄᎳ, ᎯᎠ ᎢᏴ creating ᎯᎠ ᎢᎬᏱ ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎾᎥᎢ 1854, Bernhard Riemann, ᏕᏕᎶᏆᏍᎩ Gauss, ᎠᏰᎲ ᎢᎬᎾᏔᏅᎯ ᎢᏗᎬᎾᏗ ᎠᏓᏃᎮᏗ ᎭᏫᎾᏗᏢ ᎦᏙᎯ-ᏓᎵᏆᎵᏍᎬ ᎠᎦᏎᏍᏙᏗ ᎯᎠ ᎢᎦᏛ (ᎣᏩᏒ-ᎢᎦᎢ ᎨᏒᎩ) ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏂᎦᏛ ᎤᏩᎾᏕᏍᎩ ᎦᏚᎢ, ᎠᎴ ᎾᎥᏂᎨᏍᏗ ᎠᏩᏛᏗ ᏄᏓᎴᎿᎥ ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎪᎯ ᏗᎦᎸᏫᏍᏓᏁᏗ Riemann ᎣᏂᏯᎨᏍᏙᏗ ᎠᏓᏁᏟᏴᏍᏗ ᏄᎬᏫᏳᏒ ᏗᎳᏏᏙᏗ ᎾᏍᎩᎾᎢ Einstein ᎤᏤᎵ ᎪᎷᏩᏛᏗ relativity.

William Blake ᎤᏤᎵ "Newton" ᎨᏒᎢ ᎬᏂᎨᏒ ᎾᏅᏁᎲ ᎤᏤᎵ ᏗᎦᏘᎸᏍᏗ ᎯᎠ 'ᏏᏴᏫ-ᎠᎪᏩᏛᏗ' ᎠᏏᎾᏍᏛ ᎢᏳᏍᏗ ᏯᏛᎿ; ᎠᎭᏂ, Isaac Newton ᎨᏒᎢ ᎤᎾᏓᏎᎮᎸᎩ ᏥᏄᏍᏗ 'ᏩᎦᎸᎳᏗᏴ ᎠᏓᏃᎮᏗ' (1795)

ᎾᏍᎩ ᏂᎦᏰᏙᎲᏊ ᎾᏍᏋ ᎪᎯᏳᏔᏅᎯ ᏚᏳᎪᏛ Ꮎ ᎯᎠ ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏥᏄᏍᏛᎩ ᎣᏍᏛ ᏥᏄᏍᏗ ᎣᏩᏒ-ᎤᏠᏱᎭ ᏥᏄᏍᏗ Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎠᎴ ᎪᎯ ᏥᏄᏍᏛᎩ ᎢᎬᏱ ᎥᎦᏔᎲᎢ ᎾᎥᎢ Beltrami ᎭᏫᎾᏗᏢ 1868. ᎬᏙᏗ ᎪᎯ, ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏥᏄᏍᏛᎩ ᎪᏢᏅ ᎾᎿ ᎢᏗᎦᏗ ᏚᏳᎪᏛ ᏄᏍᏗᏓᏅ ᎬᏙᏗ Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ.

ᎾᎯᏳᎢ ᎾᏍᎩ ᏥᏄᏍᏛᎩ ᎾᏊ ᎤᎾᏅᏛ Ꮎ ᏄᏓᎴᎿᎥ geometric theories ᎨᏒᎩ ᏚᏳᎪᏛ ᏰᎵᏊ, ᎯᎠ ᎠᏛᏛᎲᏍᎩ ᏂᎦᏰᏙᎲᏊ, "ᎦᏙ ᎤᏍᏗ ᏌᏊ ᎾᏍᎩ ᎯᎠ theories ᎨᏒᎢ ᎪᏢᎯᏐᏗ ᎾᏍᎩᎾᎢ ᎠᏆᏤᎵ ᏄᎵᏂᎬᎬ ᎤᏜᏅᏛ?" ᎯᎠ ᏚᏳᎪᏛ ᏗᎦᎸᏫᏍᏓᏁᏗ ᎤᏍᏆᏂᎪᏗ Ꮎ ᎪᎯ ᎠᏛᏛᎲᏍᎩ ᎠᏎ ᎾᏍᏋ ᎦᏬᎯᎵᏴᏓ ᎾᎥᎢ ᏄᎵᏂᎬᎬ experimentation, ᎾᏍᎩ ᏂᎨᏒᎾ ᏚᏳᎪᏛ ᎤᏂᎦᏛᎲᏍᎩ, ᎠᎴ uncovered ᎯᎠ ᎠᏓᏅᏖᏗ ᎦᏙᏃ ᎯᎠ experimentation ᎠᏎ ᎠᏓᏠᏯᏍᏙᏗ ᎤᎪᏗᏗ (interstellar, ᎾᏍᎩ ᏂᎨᏒᎾ ᎡᎶᎯ-ᏩᎦᏛ) ᎾᎿ ᎢᏴᎢ. ᎬᏙᏗ ᎯᎠ ᏚᏙᎳᏩᏛᎲ relativity ᎪᎷᏩᏛᏗ ᎭᏫᎾᏗᏢ ᎢᏳᏍᏗ ᏯᏛᎿ, ᎪᎯ ᎠᏛᏛᎲᏍᎩ ᎠᏓᏁᏟᏴᏍᏗ ᎤᎪᏗᏗ ᎤᏟ ᎢᎦᎢ ᏓᎧᏁᎲ.

ᎾᏛᏁᎸ ᏚᏳᎪᏛ ᎤᏴᏢ

ᏂᎦᏛ ᎯᎠ ᏗᎦᎸᏫᏍᏓᏁᏗ ᎪᎱᏍᏗ ᎠᎾᏓᏛᏂ ᎯᎠ ᏚᏦᏔᏩᏘ ᎧᏃᎮᎸᏗ ᎤᏍᏆᏂᎪᏗ Ꮎ ᎾᏍᎩ ᏥᏄᏍᏛᎩ ᎾᎥᏂᎨᏍᏙᏗ ᎤᏦᏍᏗ ᎾᏍᎩᎾᎢ ᎠᏓᏃᎮᏗ ᏧᏓᎴᎿᎢ ᎤᏤᎵ ᎧᏃᎮᎸᏗ ᎤᏂᎦᏛᎲᏍᎩ ᏂᏛᎴᏅᏓ ᎤᏤᎵ ᎤᎾᎵᎪᏒ ᎪᎵᏍᏗᏱ ᏄᎵᏂᎬᎬ ᎤᏜᏅᏛ, ᎠᎴ, ᎤᏗᏗᏢ ᎢᎦᎢ, ᎤᏍᏆᏂᎪᏗ ᎯᎠ ᎤᏓᏚᎯᏌᏘ ᏭᎵᏍᎨᏗᏴ ᎾᎾᏛᏁᎲ ᎾᏍᎩ ᎢᎬᏂᏏᏍᎩ. ᎨᏯᏔᎯ ᎠᏥᎪᎵᏰᏗ ᎠᏰᎲ uncovered ᎢᎦᏛ ᎧᏃᎮᎸᏗ inadequacies ᎭᏫᎾᏗᏢ Euclid ᎤᏤᎵ ᎤᏂᎦᏛᎲᏍᎩ, ᎠᎴ ᎢᎦᏛ unstated geometric ᏚᎵᏍᎪᎵᏴ ᎦᏙ ᎤᏍᏗ Euclid ᏱᏓᏟᎶᏍᏔᏅ ᎠᏔᏲᎸᎯ. ᎪᎯ ᎤᎾᏤᎵᏛ paralleled ᎯᎠ ᏄᏍᏗᏓᏅ ᏂᎦᎳᏍᏗᏍᎬ ᎭᏫᎾᏗᏢ ᎠᏓᏃᎮᏗ ᎠᎴ ᏗᎫᎪᏙᏗ ᎢᎬᏁᏗ ᎾᏍᎩ ᎠᏂᏰᎸᏍᎬ ᎯᎠ ᎦᏛᎬᎢ ᎢᎪᎯᏓ ᎨᏒ processes ᏯᏛᎿ ᏥᏄᏍᏗ ᎤᏓᏂᏝᏅ ᎠᎴ ᎠᏓᏅᏍᏗ. ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎾᎿᎢ ᏥᏄᏍᏛᎩ ᏧᎸᏌᏓ ᎤᏚᎳᏗ ᎾᏍᎩᎾᎢ ᎢᏤ ᎠᏫᏒᏗ ᎧᏁᎬᎢ, ᎦᏙ ᎤᏍᏗ ᏯᏓᎢᏗᏏ ᎾᏍᏋ ᎧᎵᏬᎯ, ᎠᎴ ᎦᏙ ᎤᏍᏗ ᎭᏫᎾᏗᏢ Ꮭ ᎦᎶᎯᏍᏗ relied ᎾᎿ ᏗᏓᏟᎶᏍᏔᏅ ᎢᏧᎳ ᎠᏎᎯᏍᏗ ᎠᎴ ᎾᎿ ᎠᏆᏤᎵ ᎣᏓᏅᏛ ᎤᏜᏅᏛ. ᏯᏛᎿ ᎧᏁᎬᎢ ᎨᏒᎩ ᎠᏓᏁᎸ ᎾᎥᎢ David Hilbert ᎭᏫᎾᏗᏢ 1894 ᎭᏫᎾᏗᏢ ᎤᏤᎵ ᎧᏃᎮᏍᎩ Grundlagen der Geometrie (ᎠᏓᎴᏂᏍᎬ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ). ᎢᎦᏛ ᏐᎢ ᎧᎵᏬᎯ ᏗᏫᏒᏗ ᎧᏁᎬᎢ ᎠᏰᎲ ᏭᏪᏙᎢ ᎠᏓᏁᎸ ᎦᏲᎵ ᏧᏕᏘᏴᏓ ᎢᎬᏱᎨᏍᏙᏗ, ᎠᎴ ᏄᏛᏁᎸ ᎾᏍᎩ ᏂᎨᏒᎾ ᎠᏥᎸ ᎪᏢᏗ Hilbert ᎤᏤᎵ ᎭᏫᎾᏗᏢ ᎠᎵᏏᏅᏙᏗ, ᎠᏣᏅᏙ, ᎠᎴ ᎠᎾᎵᏐᏈᎸᏍᎬ Euclid ᎤᏤᎵ ᎧᏁᎬᎢ.

ᏗᎫᎪᏙᏗ ᎢᎬᏁᏗ situs, ᎠᎴ topology

ᎭᏫᎾᏗᏢ ᎯᎠ ᎠᏰᎵ ᏯᏛᎾ ᎠᎺᏉᎯ-18th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ, ᎾᏍᎩ ᎠᏓᏁᏟᏴᏍᏗ ᎬᎪᏩᏛᏗ Ꮎ ᎤᏙᎯᏳ ᎠᏓᏅᏍᏗ ᏚᏳᎪᏛ ᎤᏂᎦᏛᎲᏍᎩ recurred ᎯᎳᎪ ᎢᏳ ᎤᏠᏱ ᎠᏓᏅᏖᏍᎬ ᎨᏒᎩ ᎤᎦᏎᏍᏔᏅᎩ ᎾᎿ ᎯᎠ ᏎᏍᏗ ᎠᏍᏓᏅᏅ, ᎭᏫᎾᏗᏢ ᏔᎵ ᎢᎦᎢ, ᎠᎴ ᎭᏫᎾᏗᏢ ᏦᎢ ᎢᎦᎢ. ᎯᎠ ᎢᏴ ᎯᎠ ᏂᎦᎥ ᎠᏓᏅᏖᏗ ᎠᏟᎶᏍᏗ ᎦᏙᎯ ᎤᏜᏅᏛ ᏥᏄᏍᏛᎩ ᎪᏢᏅᎯ ᎾᏍᎩ ᎢᎬᏂᏏᏍᎩ Ꮎ ᎯᎠ ᎤᏂᎦᏛᎲᏍᎩ ᏰᎵᏇ ᎾᏍᏋ ᎠᏍᏆᏛᎯ ᎭᏫᎾᏗᏢ ᎤᏟ ᎢᎦᎢ ᎤᏁᎫᏥᏛ, ᎠᎴ ᎾᎯᏳᎢ ᎢᎬᎾᏔᏅᎯ ᎤᏤᎵᏛ ᏕᎦᎸᏛ ᎧᏁᏌᎢ. ᎪᎯ ᎢᎬᎾᏗ ᏚᎾᎦᏎᏍᏛ ᎠᏓᏃᎮᏗ- ᎠᎴ ᏗᎫᎪᏙᏗ ᎢᎬᏁᏗ-ᎪᎱᏍᏗ ᎠᎾᏓᏛᏂ ᎠᏓᏅᏖᏗ ᎤᎷᏨᎩ ᎾᏍᏋ ᎤᎾᏅᏛ ᏥᏄᏍᏗ ᏗᎫᎪᏙᏗ ᎢᎬᏁᏗ situs, ᎠᎴ ᎣᏂᏯᎨᏍᏙᏗ ᏥᏄᏍᏗ topology. ᎯᎠ ᎤᎵᏍᎨᏛ ᎢᏳᏍᏗ ᎧᏃᎮᏗ ᎭᏫᎾᏗᏢ ᎪᎯ ᏠᎨᏏ ᎨᏒᎩ properties ᎤᏟ ᎢᎦᎢ ᏂᎦᎥ ᏗᏎᏍᏗ, ᏯᏛᎿ ᏥᏄᏍᏗ connectedness ᎠᎴ boundaries, ᎤᏟ ᎬᏰᎸᏗ ᎬᎾᏬᏍᎬ properties ᎾᏍᎩᏯᎢ ᎠᏍᏆᎨᏂ, ᎠᎴ ᏚᏳᎪᏛ ᎢᏗᎦᏘᎭ ᏂᎦᏅᎯᏒ ᎠᎴ ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᏗᏟᎶᏛ, ᎦᏙ ᎤᏍᏗ ᎠᏰᎲ ᏭᏪᏙᎢ ᎯᎠ ᎠᏰᎵ Euclidean ᎠᎴ ᎬᏙᏗ-Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. Topology ᏄᎵᏍᏛ ᎠᏓᏁᏟᏴᏍᏗ ᏧᏓᎴᎿᎢ ᏠᎨᏏ ᎠᏂᏯᏩᏍᎩ ᏄᎬᏫᏳᏒ ᏭᎵᏍᎨᏗᏴ, ᎤᏟ ᎬᏰᎸᏗ ᎬᎾᏬᏍᎬ ᎠᏂᎦᏲᎵ-ᏠᎨᏏ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᎴ ᏗᎫᎪᏙᏗ ᎢᎬᏁᏗ.

ᎯᎠ 20th ᏍᏉᎯᏧᏈ ᏧᏕᏘᏴᏗ

ᏚᎾᏙᎷᏩᏛᎲ ᎭᏫᎾᏗᏢ ᎠᏓᏃᎮᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᏠᏯᏍᏔᏅ ᎯᎠ ᎠᎦᏎᏍᏙᏗ ᎠᏓᏲᎲ ᎠᎴ ᎦᏚᎢ ᎦᏬᎯᎸᏙᏗ ᏎᏍᏗ ᏠᎨᏏ, ᎤᏟ ᎬᏰᎸᏗ ᎬᎾᏬᏍᎬ ᎯᎠ ᎤᏙᎯᏳ ᎾᏍᎩ ᎠᎴ ᏂᎦᏛ ᏗᏎᏍᏗ. ᏎᏍᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎤᏩᏌᏊ, ᎯᎠ ᎠᎦᏎᏍᏙᏗ ᎤᏜᏅᏛ ᎬᏙᏗ ᎾᏍᎩ ᎤᏩᏒ ᏎᏍᏗ ᎤᎪᏗᏗ ᏗᎪᏍᏓᏱ, ᎠᏩᏛᏗ ᏗᏔᏲᏍᏙᏗ ᎭᏫᎾᏗᏢ ᎠᏰᎵ ᎪᎷᏩᏛᏗ ᎠᎴ ᎠᏓᎸᎾᏍᏗ. ᎾᏍᎩᎾᎢ ᎢᎦᏛ properties ᏌᏊ ᎯᎠ ᎠᏂᎦᏲᎵ ᏎᏍᏗ ᎤᏜᏅᏛ, ᎯᎠ ᏦᎢ-dimensional projective ᎤᏜᏅᏛ ᎦᏬᎯᎸᏙᏗ ᎯᎠ ᏔᎵ-ᏑᏓᎴᎩ ᏠᎨᏏ, ᎠᎪᏩᏛᏗ ᎯᎠ diamond theorem. ᎬᏙᏗ ᎯᎠ ᎠᏓᏁᏟᏴᏍᎬ ᎯᎠ ᎡᎵᏍ, ᎢᏤ ᏓᏕᏲᎲᏍᎬ ᏯᏛᎿ ᏥᏄᏍᏗ computational ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᎴ digital ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏗᏓᏅᏓᏁᏗᏱ ᎬᏙᏗ geometric ᎠᏓᏃᎮᏗ, ᎠᎽᏰᎵ ᎢᏯᏓᏛᏁᎸ geometric ᎾᎯᏳ ᎢᎪᎯ, ᎠᎴ ᎾᏍᎩ ᎢᎬᏂᏏᏍᎩ ᎦᏌᏙᏯᏍᏗ.

ᎠᎪᏩᏛᏗ ᎾᏍᎩ ᎾᏍᏇ

• ᏙᎪᏩᎸ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎢᏳᏍᏗ ᎧᏃᎮᏗ
• ᎤᎵᏍᎨᏛ ᏚᏂᏃᏣᎳᏅ ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ.
• Interactive ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ software
• Flatland  ᎪᏪᎵ ᎪᏪᎳᏅᎯ ᎾᎥᎢ " ² " ᎬᏩᏚᏫᏛ ᏔᎵ ᎠᎴ ᏦᎢ-dimensional ᎤᏜᏅᏛ, ᎪᎵᏍᏗ ᎯᎠ ᎠᏓᏅᏖᏗ ᏅᎩ ᎢᎦᎢ

ᎤᏓᏎᎦᏤᏗ ᏗᏕᎬᏔᏛ

• What Is Geometry? ᎾᎾᎢ ᎠᏰᎳᏍᏗ-ᎯᎠ-ᏓᎧᏁᎲ
• Geometria online ᎪᎱᏍᏗ ᎬᏔᏂᏓᏍᏗ ᎠᏓᏃᎮᏗ ᎤᎵᏍᏕᎸᏗ, ᎦᏚᎢ ᎠᎴ ᏗᎦᏟᏌᏅ ᎪᏪᎵ ᎯᎠ ᏄᎬᏫᏳᏒ ᎦᏃᎯᎵᏙ ᎠᎴ ᎠᎧᎵᏬᎯ ᏗᏎᏍᏗ, ᏗᎬᏩᎶᏒ ᎠᏓᏎᎮᏗ ᎠᎴ ᎠᎷᏚᎧᏘ ᏎᏍᏗ.
• Cabri Software ᎾᏍᎩᎾᎢ ᎠᎾᏕᎶᏆᏍᎬ ᎠᎴ ᏓᏕᏲᎲᏍᎬ ᏗᏎᏍᏗ ᎤᎬᏩᎵ ᎠᎴ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ. ᎯᎠ ᏰᎵᏊ ᏗᏙᎳᎩ ᎭᏫᎾᏗᏢ ᎥᎦᏔᎲᎢ.
• Geogebra ᎠᏎᏊᎢ ᎤᎵᏍᏓ ᏗᎫᎪᏙᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎪᎱᏍᏗ ᎬᏔᏂᏓᏍᏗ, ᎤᎵᏍᎨᏓ ᎾᏍᎩᎾᎢ exploring ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ.
• Geops ᎠᏎᏊᎢ software ᎾᏍᎩᎾᎢ ᎢᏯᏛᏁᏗ ᎠᎳᏂ ᎠᎦᏓᏗ ᎠᎴ straightedge ᎠᏁᏍᎨᎲ ᎭᏫᎾᏗᏢ ᎯᎠ ᏄᏍᏛᎢ ᎯᎠ ᎯᎸᎯᏳᎢ ᎠᎪᎢ.
• Geometry Step by Step from the Land of the Incas ᎾᎥᎢ Antonio Gutierrez.
• Geometry ᎾᎾᎢ ᎠᏰᎳᏍᏗ-ᎯᎠ-ᏓᎧᏁᎲ
• ᎢᏍᎳᎻᎩ Geometry
• Stanford ᎥᎦᏔᎲᎢ ᎤᏬᎳᏨᎯ:
  • Finitism in Geometry
  • Geometry in the 19th Century
• Online Interactive Geometric Objects ᎾᎥᎢ Elmer G. Wiens
• Arabic mathematics : forgotten brilliance?
• The Geometry Junkyard
• Geometry problems at MathWiki online resource ᎾᏍᎩᎾᎢ ᏗᎦᏎᏍᏙᏗ
• Geometry lessons in PowerPoint ᏂᎦᏛ ᎤᎾᏕᎶᏆᏍᏗ ᎬᏂᎨᏒ ᎢᏯᏓᏛᏁᏗ ᏚᏳᎪᏛ ᎠᏓᏅᏖᏗ, ᎠᎳᏍᎬᏓ ᎾᎥᎢ ᎠᎳᏍᎬᏓ, ᎬᏙᏗ ᎥᎴᏂᏙᎲ ᏓᎵᏍᏛ, ᏗᎪᏍᏓᏱ, ᎤᎵᏍᏕᎸᏗ ᎠᎴ ᏗᏎᏍᏗ ᎭᏫᎾᏗᏢ ᏂᎦᎥ. ᎪᎷᏩᏛᏓ ᏗᎦᏎᏍᏙᏗ ᎨᏒᎢ ᎾᏍᎩ ᎾᏍᏇ ᎠᏓᏁᎸ ᎠᎳᏍᎬᏓ ᎾᎥᎢ ᎠᎳᏍᎬᏓ. ᏧᏓᎴᏅᏓ ᏗᎧᏃᏗ ᎠᎴ ᎬᏔᏅᎯ ᎠᏓᏁᏗ ᎧᏁᎢᏍᏙᏗ ᎠᎴ ᎦᏬᎯᎵᏴᏛ ᎠᏍᏓᏩᏛᏍᏗ ᎯᎠ ᎠᏓᏅᏖᏗ ᎠᎴ ᎯᎠ ᎪᎷᏩᏛᏓ ᎯᎠ ᏗᎦᏎᏍᏙᏗ.