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Saturday, June 6, 2015 Category : , , , , Off

Unusual Properties of Acute Triangles

http://www.egyptorigins.org/

I list here some of the unusual properties of acute triangles. This will permit us to gain insight into why the ancient Egyptians displayed half-angles in their constructions. (Acute triangles are those in which the angles of the triangle are all less than ninety degrees. If an angle is larger than ninety degrees the triangle becomes obtuse.)
If we take the angle bisectors in any acute triangle as shown above, these are the half-angles. We can see that if these are projected across the triangle from the three vertices they all are concurrent, that is, they meet at a common point within the triangle.
Similarly, if we project the side bisectors of the triangle from the respective vertices we see that they also are concurrent.
Third, if we project the perpendiculars of the triangle from the respective vertices we see that they too are concurrent.
But this is not the end of the unusual properties of acute triangles. Many others exist.
Take the side bisector and the angle bisector shown in the drawing above, denoted by the red and blue lines respectively. Then create another line with an angle between it and the angle bisector the same as the angle between the side and angle bisectors. This is shown by the green line. Do this for the three vertices. One will find that the three new lines will also intersect at a common point. These lines are called the symmedians, and the point at which they are concurrent is called the Lemoine Point.
For more on Emile Michel Hyacinthe Lemoine (Nov 22,1840 - Dec 21,1912) see
One can find the Lemoine point by a different construction. If one draws a line from the mid-point of a side of the triangle to the mid-point of the perpendicular to that side, and thus for all three sides, the three new lines will be concurrent at the Lemoine point.
Lemoine's work in mathematics was mainly on geometry. He founded a new study of the properties of a triangle. In a paper of 1873 he studied the point of intersection of the symmedians of a triangle. He had been a founder member of the Association Française pour l'Avancement des Sciences and it was at a meeting of the Association in 1873 in Lyon that he presented his work on the symmedians.
A symmedian of a triangle from vertex A is obtained by reflecting the median from A in the bisector of the angle A. He proved that the symmedians are concurrent, the point where they meet now being called the Lemoine point. Among other results on symmedians in Lemoine's 1873 paper he showed that the symmedian from the vertex A cuts the side BCof the triangle in the ratio of the squares of the sides AC and AB. He also proved that if parallels are drawn through the Lemoine point parallel to the three sides of the triangle then the six points lie on a circle, now called the Lemoine circle. Its centre is at the mid-point of the line joining the Lemoine point to the circumcentre of the triangle.
For example, I measured the lengths of the sides in my original of the Symmedian drawing above. These were 7.81 inches for the left side, 5.09 inches for the right side, 4.91 inches from the left vertex to the crossing of the symmedian line with the base, and 2.09 inches from the crossing of the symmedian line on the base to the right vertex. The first numbers square to 60.996 and 25.908 respectively. The ratio of these two numbers is 2.35. The ratio of 4.91 to 2.09 is also 2.35, demonstrating the finding of Lemoine.
This is a diagram of the Lemoine circle. The green lines are the symmedians. The blue lines are the lines parallel to the respective sides, drawn through the Lemoine point. One can see how the Lemoine circle intersects where the parallel blue lines cross the three sides of the triangle. Since the blue lines are parallel to a side, they must cross the alternate triangle sides. That a perfect circle can be drawn through the six points seems uncanny.
Here I show the circumcircle of the triangle, with the Lemoine Circle Center. One can see that the Lemoine Circle Center is half the distance from the Lemoine Point to the Center of the Circumcircle. I do not offer a geometric proof of this fact, but from my measurements with QuickCad software I know this fact to be true.
Nearly a hundred years ago when Lemoine made his discoveries he had to do so through construction with ruler and compass. Computer power was not available to him. Many other interesting (to mathematicians) relationships exist within acute triangles. These relationships are now explored with computers. Clark Kimberling of Evansville University in Indiana has listed hundreds of interesting facts about acute triangles. See
 Consider the Incircle. If we take the point where the angle bisectors of an acute triangle meet (are concurrent) as the center point of a new circle and draw that circle so that it just touches one side of the triangle we find that it will just touch all three sides of the triangle. This new circle is called the Incircle. (Note that we are reverting to the angle bisectors to do this construction.) The perpendicular distances from the concurrent point to each side are equal.
Proof of various properties of the Incircle have been given by Ira Fine and Tom Osler of Rowan University at Glassboro, New Jersey. See their Paper. Please note that their paper is in PDF format. You will need Adobe PDF reader to view it. I post that paper with their permission.
If one examines the drawing of the Incircle, one can see how new equal triangles are formed at each vertex, with a common base determined by the (half) angle bisector. For example, the distance from the vertex A to point a is the same as the distance from the vertex A to point c. Since the distance from the center of the Incircle to the respective perpendicular base sides are equal, two similar triangles are formed, mirror images of one another. That is, A to a = A to c, a to Incircle Center = c to Incircle Center, and A to Incircle Center is the same for both triangles. Then by flipping A-c-Incircle Center over on the A-Incircle Center axis, one would obtain two congruent triangles. Similarly for B to a = B to b, and C to c = C to b.
The side bisectors, side perpendiculars, and symmedians do not provide similar relationships; the angle bisectors are unique.
Another unique point is determined for acute triangles. This is obtained by drawing lines from the respective vertices to where the Incircle perpendiculars touch the respective sides. The three lines are concurrent at the Gergonne Point. See:
See also discussion on Triangle Centers:
For short discussion on Gergonne see:
To be continued.

Why Half Angles?

Half angles are a fundamental property of circular geometry. They have been studied since time immemorial, and discussed by many geometricians.

The easiest way to describe a half-angle is to show it in the following diagram.

The semicircle DEF contains three triangles of interest.
First, is the right triangle. Regardless of where B may fall on DEF, it will always produce a right triangle, DBF.
Second is the right angle produced by ABC. A line CB is erected perpendicular to DF that just touches the circle at B. The hypotenuse AB defines this triangle. This triangle may or may not be a Pythagorean Triangle.
Third is the right angle produced by DBC. The same perpendicular line, CB, is used as one side of this third triangle. The hypotenuse DB defines this triangle. This triangle may or may not be a Pythagorean Triangle.
The angle, BDC, is always one-half the angle, BAC, regardless of where B may fall. I now proceed to demonstrate this fact.

Thales' Theorem

In geometry, Thales' theorem (named after Thales of Miletus) states that if D, B and F are points on a circle where the line DF is a diameter of the circle, then the angle DBF is aright angle.
Proof
We use the following facts: (a) the sum of the angles in a triangle is equal to 180°, (b) the base angles of an isosceles triangle are equal, and (c) the diameter of a circle expressed in angular measure is 180°.
Let A be the center of the circle. Since AD = AB = AF, ABF and ABD are the two isosceles triangles. (Each has two sides equal, and the two equal sides of each are equal to each other. However, their angles differ.)
(Equation 1):
These conditions make the angles of the isosceles triangle ABF = b + 2a = 180°, where b is the angle at the center (in this case) and a is one of the two equal angles.
(Equation 2):
Likewise for the isosceles triangle ADB, d + 2g = 180°.
Note that d = (180° - b). This takes advantage of the fact that the diameter of a circle in angular measure is 180°.
Then using the alternate expression for d, (180° - b), and substituting into Equation 2, we obtain
(180° - b) + 2g = 180°.
Simplifying we find that b = 2g.
Substituting back into Equation 1 we see that 2g + 2a = 180°.
Dividing by 2:  g + a = 90°.
Q.E.D.
Corollary 1
Equation 2 above shows that b = 2g. This means that g is the half-angle of b. This is true regardless of where B may lie on the circle.
Corollary 2
From Equation 1 we know that 2a  = (180°  - b). But (180°  - b)  is the alternate angle to b. Hence, a  = (180°  - b)/2 = d/2. This means that a is the half-angle of d. This is true regardless of where B may lie on the circle.

Converse

I offer the following without proof.
The converse of Thales' theorem is also true. It states that if you have a right triangle and construct a circle with the triangle's hypotenuse as diameter, then the third vertex of the triangle will lie on the circle.
The theorem and its converse can be expressed as follows:
The center of the circumcircle of a triangle lies on one of the triangle's sides if and only if the triangle is a right triangle.
The circumcircle is that circle that totally encloses the triangle. For a right triangle the diameter is the hypotenuse.

History

Thales was not the first to discover this theorem since the Egyptians and Babylonians must have known of this relationship. However surviving evidence does not provide evidence that they could prove the theorem, and the theorem is named after Thales because he was said to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are equal, and that the sum of angles in a triangle is equal to 180°
As I have shown, the knowledge of the ancients was far more than empirical. Through the remnants of the mathematical documents available to us we know their level of knowledge was far greater than indicated in the Greek evidence.

Greek Proof

The Greek mathematician, Euclid of Alexandria, (c 325 - c 265 BC), in his Proposition 20, Book 3, gave a general proof that the half-angle statement is true. (Euclid labeled his points differently.)

The numbers 1.5 and 1.32 refer to other Euclid Propositions which serve to prove the missing part, and which you may find atEuclid Prop attached.

More Unusual Properties of Acute Triangles

The Nine-Point Circle of Triangle ABC with orthocenter H passes through:
1. the midpoints L, M, and N of the three sides,
2. the feet of the altitudes D, E, and F to those sides, and
3. the points X, Y, and Z, which are the midpoints of the segments AH, BH, and CH, respectively.





1. The Nine-Point Center U lies on the Euler Line of Triangle ABC.
The Euler line is the line passing through
2. the orthocenter H,
3. the circumcenter CC, and
4. the centroid G of a triangle.





The tangents to the Nine-Point Circle at the midpoints L, M, and N of the sides of the triangle form a triangle, RST, that is similar to the orthic triangle (the triangle DEF). In fact, the sides of this triangle are parallel to those of triangle DEF.
For an extensive listing of 20 additional properties about the nine point circle, see the following reference, specifically pages 53-56:
MacKay, J. S. (1892). History of the Nine Point Circle. Proceedings of the Edinburgh Mathematical Society, (11). pages 19-61.







Feuerbach's Theorem

The Nine-Point Circle of a triangle "touches" the incircle and the three excircles.

Altitude

In geometry, an altitude of a triangle is a straight line beginning at a vertex and ending perpendicular to (i.e. forming a right angle with) the opposite side, or an extension of the opposite side. The intersection between the (extended) side and the altitude line is called the foot of the altitude. This side is called the base of the altitude. The length of the altitude is the distance between the base and the vertex.
Corollaries
In an isosceles triangle (a triangle with two equal sides), the altitude having as base the third side will have the midpoint of that side as foot.
In a right triangle, the altitude with the hypotenuse as base divides the hypotenuse into two lengthsp and q. If we denote the length of the altitude by h, we then have the relation h2 = pq.
The three altitudes of a triangle intersect in a single point, called the orthocenter. The orthocenter lies inside the triangle (and consequently the feet of the altitudes all fall on the triangle) if and only if the triangle is not obtuse (i.e. does not have an angle bigger than a right one).
Four points in the plane such that one of them is the orthocenter of the triangle formed by the other three are called an orthocentric system.

Centroid

In geometry, the centroid or barycenter of an object X in n-dimensional space is the intersection of all hyperplanes that divide Xinto two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of X.
In physics, the centroid can, under some circunstances, coincide with an object's center of mass and also with it's center of gravity. In some cases this leads to the usage of those terms interchangingly. For a centroid to coincide with the center of mass, the object should have uniform density or the matter's distribution through the object should have certain properties, such assymmetry. For a centroid to coincide with the center of gravity, the centroid must coincide with the object's center of mass and the object must be under the influence of a uniform gravitational field.
Note that a figure's centroid need not necessarily lie within it; the centroid of a crescent, for example, lies somewhere in the central void.
The lines extending from each vertex of a triangle to the mid-point of the opposite sides are called medians.
The three medians of a triangle intersect at a common point. This is called the centroid of the triangle.
This point is also the triangle's center of mass, if the triangle is made from a uniform sheet of material

Symmedian

In geometry, three special lines are associated with every triangle, the triangle's symmedians. One starts with a median of the triangle (a line connecting one vertex with the midpoint of the opposite side) and reflects it at the corresponding angle bisector (the line through the same vertex that divides the angle of the triangle there in two equal parts). The resulting line is a symmedian. The three symmedians intersect in a single point, the triangle's symmedian point or Lemoine point.
Note that all three sets of lines, the median, the angle bisectors, and the symmedian are respectively concurrent.
A triangle with medians (blue), angle bisectors (green) and symmedians (red). The symmedians intersect in the Lemoine point L.
The symmedian point of a triangle with sides a, b and c has homogeneous trilinear coordinates [a : b : c].
The symmedian point L can also be constructed differently: the three lines joining the midpoint of a side to the midpoint of the altitude on that side intersect in L. The symmedian point of a right triangle is therefore the midpoint of the altitude on the hypotenuse.
The Gergonne point of a triangle is the same as the symmedian point of the triangle's contact triangle.

Circumcircles of Triangles

The circumcircle of a triangle is the unique circle on which all of its three vertices lie. The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors. (A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and is erected from that side at its midpoint.) This is because the circumcenter is equidistant from any pair of the triangle's points. All points on the perpendicular bisectors are equidistant from the respective points of the triangle.
 
A triangle is acute (all angles smaller than a right angle) if the circumcenter lies inside the triangle. The triangle is obtuse (has an angle bigger than a right one) if the circumcenter lies outside the triangle. The triangle is a right triangle if the circumcenter lies on the hypotenuse. This is one form of Thales' theorem.
The diameter of the circumcircle can be computed as the length of any side of the triangle, divided by the sine of the opposite angle. (As a consequence of the law of sines, it doesn't matter which side is taken: the result will be the same.) The triangle's nine point circle has half the diameter of the circumcircle.
The circumcenter always lies on one line with the triangle's centroid and orthocenter. This line is known as Euler's line.

Aethiopia and India

Sunday, March 29, 2015 Category : Off

Aethiopia (Greek Αἰθιοπία) first appears as a geographical term in classical sources in reference to the Upper Nile region, as well as all certain areas south of the Sahara desert and south of the Atlantic Ocean. Its earliest mention is in the works of Homer: twice in the Iliad,[1] and three times in the Odyssey.[2] The Greek historian Herodotus specifically uses it to refer to such parts of Sub-Saharan Africa as were then known within the inhabitable world.[3]
In classical antiquity, Africa (or Libya) referred to what is now known as Northwest Africa (Maghreb) and south of the Libyan desert and Western Sahara, including all desert land west of the southern Nile river . Geographical knowledge of the continent gradually grew, with the 1st century AD Greek travelogue the Periplus of the Erythraean Sea describing areas as far south as Zimbabwe. Αἰθίοψ (Aithiops), meaning "burnt-face", was used as a vague term for dark-skinned populations since the time of Homer.[4][5] It was applied to such dark-skinned populations as came within the range of observation of the ancient geographers i.e. primarily in what was then Nubia, and with the expansion of geographical knowledge, successively extended to certain other areas below the Sahara.

Before Herodotus[edit]

Homer (c. 8th century BC) is the first to mention "Aethiopians" (Αἰθίοπες, Αἰθιοπῆες); he mentions that they are to be found at the east and west extremities of the world, divided by the sea into "eastern" (at the sunrise) and "western" (at the sunset). Hesiod (c. 8th century BC) speaks of Memnon as the "king of Aethiopia".
In 515 BC, Scylax of Caryanda, on orders from Darius the Great of Persia, sailed along the Indus River, Indian Ocean and Red Sea, circumnavigating the Arabian peninsula. He mentioned Aethiopians, but his writings on them have not survived. Hecataeus of Miletus (c. 500 BC) is also said to have written a book about Aethiopia, but his writing is now known only through quotations from later authors. He stated that Aethiopia was located to the east of the Nile, as far as the Red Sea and Indian Ocean; he is also quoted as relating a myth that the Skiapods ("Shade feet") lived there, whose feet were supposedly large enough to serve as shade.
The Macrobians (Μακροβίοι), meaning long-lived, were a legendary tribe of Aethiopia: Fountain of Water

In Herodotus[edit]

In his Histories (c. 440 BC) Herodotus presents some of the most ancient and detailed information about "Aethiopia".[3] He relates that he personally traveled up the Nile to the border of Egypt as far as Elephantine Island(modern Aswan); in his view, "Aethiopia" is all of the inhabited land found to the south of Egypt, beginning at Elephantine. He describes a capital at Meroe, adding that the only deities worshipped there were Zeus (Amun) and Dionysus (Osiris). He relates that in the reign of Pharaoh Psamtik I (c. 650 BC), many Egyptian soldiers deserted their country and settled amidst the Aethiopians. He further wrote that of Egypt's 330 Pharaohs, 18 "Aethiopian" Pharaohs before the Sabacos Kushite dynasty (i.e. the Memnon or 18th dynasty). He asserts that Aethiopia was one of the countries that practiced circumcision.
Herodotus tells us that king Cambyses of Persia (c. 570 BC) sent spies to the Aethiopians "who dwelt in that part of Libya (Africa) which borders upon the southern sea." They found a strong and healthy people. Although Cambyses then campaigned toward their country, by not preparing enough provisions for the long march, his army completely failed and returned quickly.
In Book 3, Herodotus defines "Aethiopia" as the farthest region of "Libya" (i.e. Africa): "Where the south declines towards the setting sun lies the country called Aethiopia, the last inhabited land in that direction. There gold is obtained in great plenty, huge elephants abound, with wild trees of all sorts, and ebony; and the men are taller, handsomer, and longer lived than anywhere else."[6]

Other Greco-Roman historians[edit]

The Egyptian priest Manetho (c. 300 BC) listed Egypt's Kushite (25th) dynasty, calling it the "Aethiopian dynasty". Moreover, when the Hebrew Bible was translated into Greek (c. 200 BC), the Hebrew appellation "Kush, Kushite" became in Greek "Aethiopia, Aethiopians", appearing as "Ethiopia, Ethiopians" in the English King James Version.
Agatharchides provides a relatively detailed description of the gold mining system of Aethiopia. His text was copied almost verbatim by virtually all subsequent ancient writers on the area, including Diodorus Siculus and Photius.[7]
With regard to the Ethiopians, Strabo indicates that "those who are in Asia, and those who are in Africa, do not differ from each other."[8] Pliny in turn asserts that the place-name "Aethiopia" was derived from one "Aethiop, a son of Vulcan"[8] [the smith-god Hephaestus[9]]. He also writes that the "Queen of the Ethiopians" bore the title Candace, and avers that the Ethiopians had conquered ancient Syria and the Mediterranean. Following Strabo, the Greco-Roman historian Eusebius notes that the Ethiopians had emigrated into the Red Sea area from the Indus Valley, and that there were no people in the region by that name prior to their arrival.[8]
The 1st century AD Greek travelogue the Periplus of the Erythraean Sea first describes the Horn of Africa littoral, based on its author's intimate knowledge of the area. The Periplus does not mention any dark-skinned "Ethiopians" among the area's inhabitants. They only later appear in Ptolemy's Geographia, but in a region far south, around the "Bantu nucleus" of northern Mozambique. According to John Donnelly Fage, these early Greek documents altogether suggest that the original inhabitants of the Azania coast, the "Azanians", were of the same ancestral stock as the Afro-Asiatic populations to the north of them along the Red Sea. Subsequently, by the 10th century AD, these original "Azanians" had been replaced by early waves of Bantu settlers.[5]

Greek and medieval literature[edit]

Several notable personalities in Greek and medieval literature were identified as Aethiopian, including several rulers, male and female: Memnon and his brother Emathion, King of Arabia. Cepheus and Cassiopeia, parents of Andromeda, were named as king and queen of Aethiopia. Homer in his description of the Trojan War mentions several other Aethiopians. Ptolemy the geographer and other ancient Greek commentators believed that the "Aethiopian Olympus" was where the gods lived when they were not in Greece.

See also[edit]

References[edit]

  1. Jump up^ Homer Iliad I.423; XXIII.206.
  2. Jump up^ Homer Odyssey I.22-23; IV.84; V.282-7.
  3. ^ Jump up to:a b For all references to Ethiopia in Herodotus, see: this list at the Perseus project.
  4. Jump up^ Αἰθίοψ in Liddell, Scott, A Greek–English Lexicon: "Αἰθίοψ , οπος, ὁ, fem. Αἰθιοπίς , ίδος, ἡ (Αἰθίοψ as fem., A.Fr.328, 329): pl. 'Αἰθιοπῆες' Il.1.423, whence nom. 'Αἰθιοπεύς' Call.Del.208: (αἴθω, ὄψ):— properly, Burnt-face, i.e. Ethiopian, negro, Hom., etc.; prov., Αἰθίοπα σμήχειν 'to wash a blackamoor white', Luc.Ind. 28." Cf. Etymologicum Genuinum s.v. Αἰθίοψ, Etymologicum Gudianum s.v.v. Αἰθίοψ. "Αἰθίοψ". Etymologicum Magnum (in Greek). Leipzig. 1818.
  5. ^ Jump up to:a b Fage, John. A History of Africa. Routledge. pp. 25–26. ISBN 1317797272. Retrieved 20 January 2015.
  6. Jump up^ Herodotus Histories III.114.
  7. Jump up^ Journal of the Royal Asiatic Society of Great Britain and Ireland. Cambridge University Press for the Royal Asiatic Society. 1892. p. 823. Retrieved 20 January 2015.
  8. ^ Jump up to:a b c Turner, Sharon (1834). The Sacred History of the World, as Displayed in the Creation and Subsequent Events to the Deluge: Attempted to be Philosophically Considered, in a Series of Letters to a Son, Volume 2. Longman. pp. 480–482. Retrieved 20 January 2015.
  9. Jump up^ Pliny the Elder Natural History VI.35. "Son of Hephaestus" was also a general Greek epithet meaning "blacksmith".

The Blameless Aethiopians

“For who ever believed in the Aethiopians before actually seeing them? or what is not deemed miraculous when first it comes into knowledge? how many things are judged impossible before they actually occur? Indeed the power and majesty of the nature of the universe at every turn lacks credence if one’s mind embraces parts of it only and not the whole.”
Pliny, Natural Histories. 
Pliny the Elder (AD 23-79), Roman historian and scientist.
HerodotusIn Ancient times, Africans in general were called the Aethiopians1.The word Aithiops was the Greek word Herodotus and the ancients used to describe all known lands in Africa, South of Egypt.
Homer (c800BC) was the first Greek writer to mention the Aethiopians. He mentions them in both the Iliad and the Odyssey. To the ancient Greeks and early Romans, all of Africa consisted of Egypt and Aethiopians peopled of course by Black Africans. It was to the land of the “blameless Aethiopians” that Homer’s gods led by Zeus himself, repaired to feast for twelve days.
Pliny the Elder described Adulis, which port he said was the Aethiopians’ principal trading town. He stated that the term “Aethiopia” was derived from an individual named Aethiops, said to be the son of Hephaestus (aka Vulcan)2. This etymology was followed by all authorities, until around 1600, in the age of racism, when Jacob Salianus in Tome I of his Annales first proposed an alternate hypothesis deriving it from the Greek words aitho “I burn” and ops “face”3. 
The Greek poets Hesiod (c 700 BC) and Pindar (c 450 BC) speak of Memnon as the “king of Aethiopia”, and further state that he founded the city of Susa (in Persia, Modern day Iran. ). Other Greeks, Scylax (c515BC), Hecataeus of Miletus (c 500BC) also wrote books about Africans but their works have been lost through time.
To the Ancients, Egypt was the stuff of dreams and the Aethiopians were the highest manifestation of these dreams. In their time, it was acknowledged that the root of the Egyptian civilisation was from the interior of Africa. For example, Diodorus writing about the major festival in Egypt,
“…for each year among the Egyptians the shrine of Zeus is carried across the river into Libya (Africa) and then brought back some days later, as if the god were arriving from Aethiopia (Africa)”4.
With reference to the race of the Egyptians, we have Herodotus’s eye witness accounts that categorically state the colour of their skins. IT is instructive that despite the so-called leading “experts” knowledge of this passage in The Histories, they still insist on separating Egyptian culture and the amazing pyramids from Africa. We should learn from this:
“There can be no doubt that the Colchians are an Egyptian race. My own conjectures were founded, first, on the fact that they are black-skinned and have woolly hair, which certainly amounts to but little, since several other nations are so too. But further and more especially, on the circumstance that the Colchians, the Egyptians, and the Aethiopians, are the only nations who have practised circumcision from the earliest times5.”
Herodotus KNEW they were Black. He visited Egypt in 450BC and saw them with his own eyes as opposed to modern-day experts. But as for the Aethiopians, nobody has ever argued their blackness. Of these black people, Herodotus was unstinting and verbose in his praise. In Book 3, Herodotus defines “Aethiopia” as the farthest region of Africa. He continued:
“Where the south declines towards the setting sun lies the country called Aethiopia, the last inhabited land in that direction. There gold is obtained in great plenty, huge elephants abound, with wild trees of all sorts, and ebony; and the men are taller, handsomer, and longer lived than anywhere else6.”
Ptolemy the geographer and other ancient Greek commentators believed that the “Aethiopian Olympus” was where the gods lived when they were not in Greece.
Strabo (Greek historian, geographer and philosopher. c64BC – 24AD) stated that some previous authors had considered Aethiopia’s northern border to begin at Mount Amanus, thus including all of Syria, Israel and Arabia.
The Greeks and the subsequent Romans had a healthy respect for the intelligence of Africans and respected them especially for their piety. Lucian mentions a thoughtful Negro scribe from Memphis named Pancrates. Aesop, of the Aesop’s tales was described as an Aethiopian.
Diodorus wrote that the Aethiopians were like the gods and without fault; Claudius Aelianus (Roman Author c 175AD – c235 AD) believed that the gods bathed in Aethiopia and Stobaeus (c 5th century AD) recorded that the Aethiopians do not need doors on their homes and do not steal the possessions that their neighbours leave in the street. Of no other part of the world is there a constant reference to such general honesty and lack of crime. (see the echo of this 2,000 years later in Ibn Battuta’s writing on Mali)
In the ancient Greek Romance Aethiopica by Heliodorus of Emesa (Greek writer, c 3rd century AD), an Aethiopian King was a model of wisdom, righteousness, and magnanimity:
“The king does not condemn people to death, and sends out messengers to tell his military troops not to slaughter the enemy, but to let them live when they have been defeated. The king proclaimed, “A noble thing it is to surpass an enemy in battle when he is standing, but in generosity when he has fallen.”

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