🌍 History: Geometry in India & World
"Geometry" word aaya hai Greek language se: Geo (Earth) + Metrein (Measure). Matlab "Zameen ko naapna".
Puraane zamaane mein logon ko zameen naapne ki zarurat padti thi, buildings banane ke liye, aur religious altars (havan kund) banane ke liye.
India mein Sulba Sutras (around 800 BC) geometry ka earliest written record hai. Vedic times mein, yagya ke liye jo altars bante the, unki shapes (squares, circles) bahut precise honi chahiye thi.
Example: Agar ek square altar banana hai jiska area circle altar ke barabar ho, toh iske liye geometry ke rules use hote the.
Lekin pehle geometry knowledge bikhra hua (scattered) tha. Phir aaya ek teacher — Euclid (Egypt, 300 BC) — jise "Father of Geometry" kaha jata hai. Usne us time ki saari geometry ko collect kiya aur ek systematic book likhi jiska naam tha "Elements".
🏗️ Euclid's Definitions
Euclid ne geometry ko define karna start kiya sabse choti cheezon se. Kuch examples dekho:
1. Point: Ek point wo hai jiska koi part nahi hota. (Sirf ek dot . )
2. Line: Ek line wo hai jiski sirf length hoti hai, choudai (breadth) nahi hoti.
3. Surface: Surface wo hai jiski length aur breadth dono hoti hai.
🤔 Axioms vs Postulates: Kya Fark Hai?
Euclid ne kuch baatein mani jo "Universal Truths" thi – unhe prove karne ki zarurat nahi thi. Unko usne Axioms aur Postulates kaha.
- Axioms: Aisi general sachai jo har jagah apply hoti hai (sirf Maths mein nahi).
Ex: "The whole is greater than the part." (Pura cake hamesha uske ek tukde se bada hota hai). - Postulates: Aisi sachai jo sirf Geometry ke liye bani hain.
Ex: "Hum kisi bhi do points ko jod kar ek straight line bana sakte hain." - Theorems: Ye wo statements hain jo axioms aur postulates ka use karke PROVE karni padti hain.
🖐️ Euclid's 5 Postulates
Euclid ke 5 famous postulates jo geometry ki neev (foundation) hain:
- Straight Line: Kisi bhi ek point se dusre point tak ek seedhi line (straight line) draw ki ja sakti hai.
- Terminated Line: Ek ruki hui line (line segment) ko dono taraf kitna bhi lamba kheench (produce) sakte hain.
- Circle: Kisi bhi point ko center maankar aur kisi bhi distance ko radius lekar ek circle draw kiya ja sakta hai.
- Right Angles: Saare Right Angles (90°) ek dusre ke barabar hote hain.
- Parallel Postulate: Agar ek line doosri do lines ko cross kare, aur ek taraf ke interior angles ka sum 180° se kam ho, toh wo dono lines aage jakar usi taraf milengi (intersect karengi). Iska simple matlab: "Parallel lines kabhi nahi milti."
5th Postulate thoda complicated lagta hai, par ise "Train Tracks" se samjho.
Agar train ki patriyan (tracks) ek dusre se perfectly parallel na ho, aur thodi si bhi jhuki hui ho, toh kya hoga? Kahi na kahi door jaakar wo aapas mein takra jayengi (meet kar jayengi).
Yahi 5th Postulate kehta hai: Agar lines parallel nahi hain (Matlab angle sum 180° nahi hai), toh wo meet karengi.
📊 Visual Summary: The 5 Postulates
⚖️ Relationship between Axiom and Theorem
Axiom ek basic trust hai, aur Theorem uska logical result hai.
Axiom 5.1: Given two distinct points, there is a unique line that passes through them.
(Agar do alag points A aur B hain, toh unse sirf ek hi line guzar sakti hai.)
Theorem 5.1: Two distinct lines cannot have more than one point in common.
(Agar do alag lines hain, toh wo sirf ek hi point par cross kar sakti hain. Wo baar-baar cross nahi kar sakti jab tak wo straight lines hain.)
Proof ka Logic: Agar wo 2 points par cross karti, toh unn 2 points se 2 alag lines ban jati, jo Axiom 5.1 ke khilaaf hota. Isliye, wo sirf 1 point par hi cross kar sakti hain.
📌 Aage Ka Content (Next Chapter)
Euclid ki geometry sirf shapes tak seemit nahi thi, usne humein logically sochna sikhaya. Next chapter mein hum inhi concepts ko use karke Lines aur Angles ke problems solve karenge.