Comments/Ratings for a Single Item

Using magic number = 0.7, I calculate the mobility of the Ferz->Bishop bent
rider as 8.8, placing it midway between the Crooked Bishop (mobility 9.7)
and Cylindrical Bishop (mobility 7.9)--so, roughly Rook-value, or a smidge
less.

I think only its first 3 moves are the same as the Zigzag Bishop, though,
not the first 4.

The value of Griffon and Aanca

Synergy - Betza's calculation does not take into account that in combining sliders to a compound, there usually is a synergy. Applying his method to the orthodox Queen, for example, would lead him to conclude its value was that of Rook (5) + Bishop (3-3.5), i.e. 8-8.5, while every Chess player of course knows it is more like 9.5-10. The reason, no doubt, is that the B moves of the Queen help you to position her such that the R moves can do damage, and vice versa.

Griffon - The true value of the Griffon in a FIDE context is slightly under that of Queen minus Pawn, around 8.3 (if we take the Kaufman value 9.5 for Q). That is, a Queen lightly beats a Griffon + Pawn (like 53%), if that imbalance is present in the opening position. (I.e. replace the Queen of one side by a Griffon, and delete the f-Pawn of the other side.) That is 1.66 times the Rook value, 0.2 larger than Betza's 'naive' adding of the values of the left and right Griffon. (Which was not that naive, because it had to correct for moves they had in common.) So there seems to be a synergy of about 100 cP.

Aanca - Similarly, the difference of the Queen and Aanca turns out to be significantly less than that of Knight and Pawn: Aanca + Knight beat Queen + Pawn, from the FIDE opening position (i.e. delete a Knight for one side, replace the Queen by Aanca for the other and delete his f-Pawn, and the Aanca scores ~59%, which correspond to about half a Pawn. As Knight minus Pawn is about 2-2.5, this means Queen minus Aanca ~ 1.5-2. That makes Aanca something like 7.5-8, say 7.8. Which is 2.4 times the Bishop value if we set the latter at 3.25 (the Kaufman value of a lone Bishop) and 2.22 x B is we set it at 3.5 (to incorporate half the B-pair bonus of 0.5 Pawn). In any case way larger than what Betza calculates.

The Chiral Bent Riders

Chiral Aancas - I was of course curious why the Aanca value would be so much above Betza's guesstimate. So I decided to check the individual steps. The first step is to get the value of the 'left' or 'right' Aanca, which Betza values as "8/7 of a Bishop plus some bonus for not being color bound". So I set up a match where one side has the Bishops replaced by a Left and a Right Aanca. I put them such that they bend 'inward', i.e. white's Left Aanca starts on f1, so that it can move f1-f2-e3-d4-c5-g6 as soon as the f-Pawn is pushed. When black has the Chiral Aancas, he starts with the Right Aanca on the f-file, as FIDE is mirror symmetric, and the chirality flips on reflection.

Turns out the Chiral Aancas win by about 60%, which is over half a Pawn, even if you start them with Pawn odds. (In which case I delete the g-Pawn in stead of the regular f-Pawn, because the latter would develop the Aanca like normally a missing g-Pawn would develop the Bishop.) This means a pair of Chiral Aancas is worth about 150cP more than a pair of Bishops, meaning that a single Chiral Aanca is a full Pawn more valuable than a lone Bishop (or Knight). With the Kaufman values B = N = 325, so that a Chiral Aanca would be 425cP.

That makes the high value of the complete Aanca understandable: adding the values of Left and Right would produce 850, but you have to subtract ~150 for the overlapping W moves, which would leave you with 700. Now I found about 780, and a synergy of 80 cP sounds reasonable, although a bit on the low side when we realize a Queen, which is not that more valuable in total, has ~150cp synergy between R and B moves.

Chiral Griffons - I did a similar test on the Griffon, where I started one side with Chiral Griffons in stead of Rooks (such that the one on a1 would bend left: a1-b2-b3-b4-...). Because it was too cumbersome to set it up such that Fairy-Max could castle with the Griffons if I gave them alternately to black and white, I also forbade castling with the Rooks.

The Rooks won this match very lightly, some 54%, translating to ~1/8 Pawn value loss per Chiral Griffon compared to Rook. Now Rooks and other Wazir-like pieces are known to test about 25cP low from opening positions compared to other pieces (i.e. they test as 475cP rather than the classical 500). They only reach their full potential once there are open files (or, in the case of Wazir, managed to get in front of the Pawns). Now I am not sure if the Chiral Griffons suffer a similar devaluation in the opening. According to the test the opening value would be ~465cP. If they do their value on a non-crowded board might rise to 490cP. For comparison, Betza's 7/8 of a Rook would amount to 440cP even when you set R=500, and seems a slight under-estimate.

Combining the two Chiral Griffons into a complete Griffon would then naively give 2 x 465 = 930cP, of which after subtracting a penealty of ~150 (The Ferz value) for the moves they had in common 780 would be left. The empirical value of 830 then includes 50cP worth of synergy. If we assumed 490cP for the Chiral Griffons, there would not be any synergy at all. (Which does seem unlikely. So Chiral Griffons are probably not really dependent on open files for getting them into play. Which fits with the observation that you can develop them through Pg2-g3, RGh1-g2, after which the Right Griffon already covers the f-file and the back rank.)

Mating potential - As a side remark: the Chiral Griffons have the capability to force checkmate on a bare King.

Just a thought - By playing with these pieces by hand on WinBoard, which highlights destination squares, I noticed that the Aanca has the same move pattern as two Bishops standing left and right of it (or, alternatively, in front of and behind it). The moves of these 'virtual Bishops' intersect each other in two squares, but the Aanca compensates this by also covering the two squares these virtual Bishops occupy, which the Bishops would not. So it does seem an Aanca should be the equivalent of two complete Bishops. The situation with the Griffon is different, though: most of the move pattern of a Griffon on e4 would be reproduced by a pair of Rooks on d3 and f5 (or d5 and f3). These would also intersect in two squares, and have the Griffon attack the squares they are on. But all W squares would not be covered by the Griffons, while they would be covered by the Rooks. So the naive Griffon value should be appreciably smaller than double the Rook's, unlike the Aanca.

For other newcomers than Florea, here are Gryphon and Aanca whose values Muller and Aurelian Florea have pinpointed through fascinating Fairy-Max discussion. It makes sense to player that preferred-emphasis-orthogonal Gryphon edges out Aanca. But are not G. and A. better implemented on 10x10 or even up to 12x12? Then on large boards they might equal Queen. In any case on 8x8 and 8x10 Gryphon and Aanca are higher value than 5.0-range Rook and Falcon. Of the four fundamental Chess pieces Bishop and Knight as 3.0 and Rook and Falcon as 5.0 are pairwise very close in values given the standard boards 8-deep and reasonably popular piece mixes. Betza had just become aware of Falcon when writing up "Bent Riders" in 2002, and Falcon's changing direction has the B.R. piece genre's mode but that move just beyond Knight of Falcon is better described as fixed-length plural-path.

The Florea-Muller exchanges on Fairy-Max are reminiscent of those of Reinhard Schamagl (Capablanca Random) ten years ago on Carrera-Capa RN and BN.

Another idea that I have would be knight then camelrider

@Kelvin that thought has crossed my mind too. But it had been any camelrider move after the knight move. Works with zebras, too I guess.

The opposite version and the zebra pieces. I will edit them later with the correct piece picture

One can also make many other Rook/Bishop-based bent sliders than Griffon and Aanca. The latter two bend their trajectories by 45 degrees after one step, but one could also bend by 90, 135 or even 180 degrees, and after another number of steps. Last year someone presented me a variant he designed, which featured a piece that moved orthogobally for maximally 10 steps, turning at right angles after exactly 5 steps.

@HG

Remember your point for an upcomming game of mine!...

I came on this page and I am horrified to read what I read.

"In H.J.R. Murray's History of Chess, page 181 states that the Alfonso manuscript was published in about the year 1211" >> no, not 1211! Murray wrote it right: 1283. 

"which on page 346 is said to have used algebraic notation, and to have described a chess variant that included the modern B and Q": not at all!
That chess variant wich used algerbric notation and modern move is another one, from India, written in Persian and dated 1796-8. It is reported quite clear in Murray page 181 for who has eyes to read!

Page 348, Murray gave a short description of Grant Acedrex from King Alfonso X.Today this is better known thanks to the PhD work of Sonja Musser. I worked a bit with her on this, this is reported in my book A World of Chess (McFarland, 2017). In few words: what was called Unicornio in medieval spanish was clearly a Rhinoceros. So the Rhino was a piece first jumping like a Knight, then going away like a Bishop.

It was the counterpart of another piece moving one step diagonal then moving away on rows and columns. That later piece is called Aanca in the manuscript. It's an Arabic word, not Spanish, designating a giant Eagle or prey bird, from oriental legends (able to carry elephants). This was mistakenly traduced by Gryphon by Murray. This is unfortunate as the Gryphon was a very different legendary animal. This is why I prefer to use the name of Eagle in Metamachy and not Gryphon to avoid replicating that mistake.
 

"Not described there is a piece which makes a one step Rook move and then continues outwards as a Bishop. For lack of a name, I'll call it the Aanca." 
No no no please! Aanca is the Giant Eagle, or the Gryphon if you want. Do not give that name of Aanca to a piece which is different and is more like the Unicornio / Rhinoceros. This is a very very bad idea. Stop adding confusion, I wish one is more careful when reading the work of Murray.

We have to blame Ralph Betza for hijacking the name Aanca, and assign it to the bent slider that first makes a W step and then continues outward like B. Unfortunately this new meaning of Griffon / Aanca became so well settled that it will be difficult to eradicate. There was a discussion about this in the comments on Team-Mate Chess (which also features a Betza Aanca).

Note that such a confusion is not unique: the Spanish word for a Bishop is Alfil, but in English Alfil is used in the original sense of the Shatranj piece. It will be even harder to make the Spanish Chess community see the error of their ways, and make them drop their erroneous use of the word Alfil.

My conclusion was that the simplest solution is to just accept that chess men have different names in different languages, which are not always translation of each other (e.g. Bishop - Runner - Elephant - Fool - Counselor), and that this can also be the case for unorthodox pieces. That way Griffon would be the English name for the piece that the Spanish call Aanca, while Aanca would be abused in English to dscribe the Betza piece, similar to how the Spanish abuse the word Alfil to describe a Bishop.

Dear HGM, I'm sorry to disagree with your demonstration.

It is very unfair to say that the "Spanish abuse the word Alfil". This is denying chess history. First we largely owe the Spanish to have transmitted to the rest of Europe chess from the Arabs in the years 950-1000. Alfil in Arabic is/was "al fil" which means "the elephant". It stayed in Spanish as Alfil, and was adopted in France as "fol" in Occitan, then "fou" in French, meaning "fol". Due to the shape of the piece, also borrowed from the Arabs, with 2 protuberances for the tusks, it was assimilated to a bishop's miter by Englishmen. So the Spanish just kept the name. The move was only modified 500 years later, also first in Spain! It is not because some American and English chessvariant lovers decided to revive the world Alfil in the 20th century that Spanish have abused whatever. 
Consider that Russians are calling the bishop a "slon" which also means elephant in Russia! Russia for chess is something no? Spanish are not so wrong  after all. But OK, I will not say that English are abusing :=)

I would agree with you that it would be difficult to forget the Gryphon or Griffon for CV lovers, but Aanca no. Aanca is Anka in Arabic and is what I said, a legendary giant Eagle, found in The 1001 Night Tales. In context of chess, it is found in Alfonso X's codex where it has clearly the move of what you call the Griffon. And shall I need to remind that Alfonso X was not writing in modern English but in medieval Castilian. (tired to read that the world griffon is found in the Libro de los juegos from king Alfonso >> it is not!).

Actually, the same Alfonso described a piece, in the same Grant Acedrex, almost moving like W followed by diagonals. Actually, it was just skiping the W squares, first jumping like a knight to the diagonal. That was an Unicornio, that it is demonstrated that it meant a Rhinoceros. Alfonso didn't have access to Wikipedia to check what a Rhino is. So, rather than Aanca, Rhinoceros would be a much better name. This is what I selected for Zanzibar chess. 

Calling that piece an Aanca, is like calling the a Bishop a Rook. Absurd.

I can confirm that the Russians do call the bishop an elephant "slon" (although their physical boards have pieces that look the same.)  So I would agree that the Spanish are in the mainstream by calling it an alfil.

I also don't think the use of "aanca" is that well established and, given this information, I think we should tend toward something else.

What to do with this page is a bigger question.  This page is more of an opinion piece by Betza than an encyclopediac entry.  (Despite the icon, this page isn't in the Piececyclopedia.)  That said, the page does present some things as facts that appear to be incorrect.  I am in favor of - at a minimum - correcting obvious errors with footnotes indicating we have updated Betza's text for accuracy.  Even better would be to retire this page and replace it with a new, more encyclopedia-worthy page on bent riders.  For example - I care not for the discussion about whether space aliens would use our chess pieces.

Betza wrote:

Not described there is a piece which makes a one step Rook move and then continues outwards as a Bishop. For lack of a name, I'll call it the Aanca (13th century Spanish for "Gryphon"). Although the Aanca is not described, one can suppose that the same mind who conceived the Gryphon and the Unicorn probably also considered the Aanca.

This is really irresponsible. Lacking a name for the Gryphon's orthogonal/diagonal counterpart, he just borrowed another name used for the piece called a Gryphon. The Aanca was described. It was described as being the same piece as the Gryphon. What wasn't described was a piece that Betza should have found a more appropriate name for. I don't have sufficient knowledge of Arabic or Spanish, but Gryphon seems like a fitting name on the basis that this piece is sort of a hybrid of Rook and Bishop, and the Gryphon is a mythological hybrid animal. With that in mind, it would be appropriate to use the name of another mythological hybrid for the corresponding piece, such as Chimera or Manticore.

Actually, I think Manticore is an excellent name.

Since the name Unicorn is already in use for other pieces, the piece he describes as moving like a Knight, then a Rook, could be called a Hippogriff, which I just learned is the offspring of a Gryphon and a mare. The hippo- root in the name means horse, which fits with a Knight move.

I used the name Aanca for apothecary chess modern and griffin too (I thiught that is the correct spelling at the time). So I don't see any problem with these. Anyway nobody can expect total consistency as one name for 1 piece over all chess variants.

Previous time this subject came up I proposed to rename the W-then-B to Ancaa.

I don't really like the name Eagle for the Griffon. And Eagle is an ortdinary bird, not a mythical monster. In that sense Griffon is much closer: a large mythical monster that can fly. OK, it doesn't really prey on Elephants, but who does? Arakis Sandworms, I suppose, but these cannot fly. If we want to be purist, we should keep the Arab name Roc.

Renaming the Grant Acedrex Unicorn is just as bad as renaming the Aanca. Unless we would rename it to Rhino, which was what the Alfonso Codex really meant. Using Unicorn or Rhino for other pieces than the N-then-B is exactly the same as hijacking the name Aanca for W-then-B.

I like the name Hippogryph, but I think it should be used for the W-then-B.

As to the Alfil: the move was changed without changing the name. Betza changed the move of the Aanca, but kept the name. Sounds like it is the same thing.

Using Unicorn or Rhino for other pieces than the N-then-B is exactly the same as hijacking the name Aanca for W-then-B.

No, it isn't, because people have used the name Unicorn for other pieces without borrowing it from another Chess variant where it was used for a different piece. Unlike Aanca, which is a word Betza came across only in Murray's description of a particular Chess variant, people far and wide know of unicorns and rhinos. This is one of the reasons I put a Unicorn piece in the logo. I knew people would recognize it even if they weren't familiar with games that use it.

I like the name Hippogryph, but I think it should be used for the W-then-B.

The name seems more appropriate for a piece with some kind of hippogonal move.

As to the Alfil: the move was changed without changing the name.

You have that backwards.  As Chess evolved the elephant was enhanced from leaping diagonally to sliding diagonally.  It was at a later point that the English name (and maybe other languages) was changed to bishop while other languages, such as Spanish and Russian retained the existing name.

As to the Alfil: the move was changed without changing the name. Betza changed the move of the Aanca, but kept the name. Sounds like it is the same thing.

Here's the difference. Players of Chess were using certain equipment with certain piece names, and while doing this, they changed some of the rules, including how some pieces moved. Ralph Betza read in a book about a game that was not played by anyone he knew. The description of the game gave Spanish and English names for the pieces. He got an idea for a piece similar to one of the pieces in the game, and he decided to name it after the Spanish name for the piece it was similar to. This would be like reading about Chinese Chess, learning about the Cannon, coming up with the idea for a diagonal moving counterpart to the Cannon, and calling it a Pao after the Chinese name for the Cannon.

Well, I suppose it depends on whether one considers it more important how the piece moves, or that it stayed in the array. The problem I see with the latter is that I consider Courier Chess as an important milestone on the evolutionary path of modern Chess, and consider the Bishop as a descendant of the Courier rather than the Alfil. But it is clear that I am outvoted here.

I still dislike Eagle for F-then-R, though. If a more bird-like name is desired, I would go for Harrier. This at least also exists in a chimeric (or should I say cyborg?) version that is able to take on Elephants...

Interesting discussion. I don't want to push for my choice of Eagle for F-then-R as Gryphon is more popular in CV community. Eagle is just my personal choice in the frame of Metamachy as it has some resonance with the Lion, carrying some heraldic meaning. Btw, my use of Lion, and also Elephant is also personal and I don't push. 
In the case of Aaca by Betza it is primarly the result of a wrong reading of Murray. There are more examples in that page that demonstrate that Betza read it very very quickly, the least to say. So, we shouldn't keep what is really a mistake.
N-then-B is exactly the Unicorn, drawn as a Rhinoceros by the medieval artist in the original codex. I don't see the need for another word, but if you believe so, why not Hippogrif. Finding another mythic animal for W-then-B is a good idea. Maybe one with a big horn.

To Greg Strong: 

you wrote: "As Chess evolved the elephant was enhanced from leaping diagonally to sliding diagonally.  It was at a later point that the English name (and maybe other languages) was changed to bishop while other languages, such as Spanish and Russian retained the existing name."

As the matter of fact, no, it didn't happen that way, they didn't change the name after the move was changed. The name was changed much before.

Old chess, similar to shatranj, was transmitted to Christian countries and reached England about 1050. The "Fil" was soon called Bishop there and in other Scandinavian lands. For example, the famous Lewis set has pieces depicting Bishops. The modern move for that piece (although it had some forerunners in Grant Acedrex or in Courrier Chess) has been adopted about 1475-1500. So, during almost 500 years the piece called Bishop in English was played like the one we call Alfil in our chessvariants.com pages.

Btw, same thing for the Queen, called a Queen and played as a Ferz during 500 years.

How about 'Acromantula', then? In the Harry Potter series this is a monstrous men-eating spider. I like the spider connection, because spiders have 8 legs, and the W-then-B moves along 8 rays.

And it keeps me happy that it also starts with 'A', as the Fairy-Max implementation of Team-Mate Chess uses this piece with A as ID, and changing that would break backward compatibility with the previously saved games.

Hmm, I'm not a fan of the spider, even a monstruous one. The link with Rhinoceros/Unicorn is too remote. We need an horn :=) But also because Musketeer Chess has proposed and is even selling a Spider already, even though I don't like at all the definition he used.
What about the Monoceros, see https://en.wikipedia.org/wiki/Monoceros_(legendary_creature) I'm pretty sure that it was never used in CVs (I'd be surprised) and there is an obvious parenthood with Unicorn.

I read the whole Harry Potter series last year, and I think I know what you're talking about, but I don't remember the name Acromantula. The piece does have a movement pattern that is suggestive of a large spider, though. Based on similar reasoning, I used the name Spider for the Alfil/Dabbabah compound in Interdependent Chess. I guess the name Acromantula is supposed to suggest an acrobatic tarantula, though that wouldn't account for the m in the middle. Arachnid would be a more familiar name for most people, though it's also pretty generic. Ultimately, the choice will be up to whoever uses it in a game.

The name of the individual spider was Aragog. But when I Googled for that name, most references immediately mention that Aragog was an Acromantula. E.g. https://en.wikibooks.org/wiki/Muggles%27_Guide_to_Harry_Potter/Characters/Aragog . Since I am sold to an 'A', it is the name I will use in Team-Mate Chess.

If you want horns, Triceratops comes to mind. But I don't see why arachnids should be discriminated against.

Naming chess pieces after Harry Potter?  yuck

My main concern with names from Harry Potter is that they are not going to be recognized as easily as names from folklore, mythology, or zoology. Although I've read Harry Potter now, I was already an adult when it came out, and I've already forgotten most of the monsters and magical creatures from it that I wasn't already familiar with.

I am not so sure about that. I had no idea what a Kirin, Wyvern or a Manticore was before I looked it up. I doubt whether many people in the street where I live would know what a Griffon is. I think it is a safe bet that more contemporary people have read Harry Potter or seen the movie than that have read the Odyssee.

I don't see any difference between Harry Potter stories or other contemporary fantasy litterature and ancient myth.

I have the same so-so feeling than Fergus about inserting Harry Potter's here. Why not my suggestion of Monoceros, it is  a fantastic beast recognized in WP, it has a link with the Unicorn. As Unicorn is the N-then-Bishop in Alfonso's description, why not Monoceros for W-then-Bishop? The difference between the 2 moves is thin, so is the difference between the 2 names.

Why not my suggestion of Monoceros, ...

Because it does not start with an 'A'.

Besides, 'Monoceros' is just Greek for Unicorn. If we use those two names for different pieces, how should the Greeks distinguish them? As there seemed to be opposition against calling the Grant Acedrex N-then-B by its original name Unicorn, because we already associate that name with another piece, I think we should make Monoceros the English name of the N-then-B. This also sounds more like the originally intended Rhinoceros (which unfortunately was also already taken).

And note that I already have a piece called Unicorn in Team-Mate Chess. I tend to use this name for an augmented (or a royal) Knight (in this case WN), because XBoard has a piece image for it, and it is one of the most knight-like pictograms there that is not actually a Knight. I don't want two different pieces in Team-Mate Chess to have names that are just each other's translation in some other language. (Even though the move patterns are rather similar, the Acromantula just being a sliding version of the leaping Unicorn.)

OK, I understand. The idea should not come from me. Btw, the ferz-then-rook has also an 8-leg move. Good luck with your spiders, starting with A or not.

Not sure why you say that. Any suggestion from you that starts with 'A' would have been more than welcome. I had to go through a quite large set of mythical beasts before I found anything with an A at all.

Griffon is the commonly used English name for the F-then-R. There is no need to change that, as the name doesn't collide with anything else. If there was an exact translation of the historical name, it could be an argument (but not enough to justify changing a well-settelled name, IMO). But that is not the case here, and Griffon seems as close as you can get. Keeping the untranslated name (as the Spanish did for Alfil, and the English for Rook), would give either Aanca if we follow the Spanish (a bad idea, in view of the confusion sowed by Betza) or the Arabic 'Roc' (which most people would probably reject for being to close to Rook).

Only the W-then-B and N-then-B are in need of new suitable names, and Monoceros seems most suitable for the N-then-B, as it is an exact translation (to Anglicized Greek) of the Spanish name.

[Edit] Considering the lack of acclaim for Acromantula: I would also be perfectly happy with a fable animal of my own design for W-then-B. E.g. an Antigryph (or Apogryph?). Which is the opposite of a Gryphon (compare Contra-Grasshopper): A lion's head and fore-legs on an eagle's body.

I like arachnid is better than acromantula.  I looked through my old D&D monster manuals and was supprised how few items begin with A.  "Apparition" is not great but not terrible.  For more conventional beasts, there is the ferocoius Ant Eater.  There is the Ankylosaurus dinosaur.  And there is a very long list here that I haven't looked through yet:

https://en.wikipedia.org/wiki/List_of_legendary_creatures_(A)

Like the Murray Lion, the Betza Unicorn may have been created through a misreading. The Unicorn described in Murray's account of Grande acedrex is not a bent rider. The description of the Gryphon says "A move compounded of one step diagonally, followed by any number straight." If the Unicorn were also a bent rider, its description should say something like, "A move compounded of a Knight leap to an empty space, followed by any number diagonally." But it doesn't say anything like that. Instead, it says "First move = Kt (but cannot capture), afterwards = modern B." This is saying that on its first move, it moves as a Knight without capturing, and on each subsequent move, it moves as a Bishop. Of course, it is possible that Murray bungled the description. But based on Murray's description, the Unicorn is not a bent rider.

I found a translation of the Alfonso document. This translation uses the name rhinoceros or rhino, and it says this,

The rhino’s move is composed of two different steps. First, like leaps like a knight. It may remain on
that square if it wishes or may also continue to any square on the diagonal(s) of that square,
maintaining its movement in a forward direction from that square.40

However, the footnote says, " Again, I thank Jean-Louis Cazaux for his help with my translation of the rhino’s move." For all I know, and he can correct me if I'm wrong, he based his understanding of the Rhino's move on Betza's understanding of the piece. Since he was just criticizing this article, maybe he got it from a source that got it from Betza. Anyway, that would put us in a vicious circle of not knowing for sure how Alfonso actually described the piece. That translation is too recent for Murray to have used. Is there a copy available of the translation or resource that Murray used to learn about this game? Or could Jean-Louis elaborate on why the original text supports this translation?

Jean-Louis will probably be able to say more on this, but new translations of the Alfonso Codex have created a sort of consensus that Murphy was indeed completely wrong, and that the 'Unicorno' piece is an N-then-B. It depends a little bit on the interpretation of 'forward', whether this means "in the direction of the promotion rank" or "outward". The manuscript also contains descriptions of the real-life animals on which the pieces are modelled. And from this it is obvious that it is intended to depict a Rhino, even though the illustrator of the manuscript obviously had never seen one in his life, and hasd no clue as to how it looks. The move is supposed to reflect the behavior of the real-life animal, "starting with a leap, and then charging forward in a straight line". That doesn't suggest a reversal of direction after the leap; no real-life Rhino would be agile enough to do that. So it is pretty certain that 'forward' meant 'outward'.

@Greg: Interesting list. One thing I noticed is that it says

Anqa (Arabian) - Legendary Huge Satanic Eagle with Human Face. sometimes can resurrect herself like phoenix did.

So accoding to this Aanca is not Spanish for the mythical Elephant-eating bird, which was called 'Roc' in Arabic, but an Arabian name itself for an entirely different beast. And this latter beast is actually very close to what in Greek / Roman mythology is called a Harpy. This suggest the correct the correct translation for Aanca is Harpy.

[Edit] I looked up the translation of the Alfonso Codex, and from the description of the Aanca is is obvious that this is the Elephant-eating bird. So so much for the accuracy of the list... This is the quote of the translation of the Unicorno description:

The rhinoceros is a very large and very strong beast with two horns – one on its forehead and
one on its nose. Its nose horn is so strong that it can spear an elephant in the gut and lift it from
the ground. The forehead horn is very sharp and cuts powerfully. This rhinoceros is as large as
an elephant and ash coloured. It has ears like a pig and when it is angry its eyes turn as red as
ruby. When it begins to run it runs far after it gives a jump like a horse and so does its piece. The
rhino’s move is composed of two different steps. First, like leaps like a knight. It may remain on
that square if it wishes or may also continue to any square on the diagonal(s) of that square,
maintaining its movement in a forward direction from that square.

I remember having looked at the original text, and although my 13th-century Spanish is just as bad as my modern Spanish, it is still enough like Latin / French / English that is is easy to verify this translation is entirely correct. I had no doubt about that at all.

What I dislike of 'Arachnid' is that it is not the name of a species, but an entire class of species, which not only contains spiders, but also scorpions and other orders of animals. Like calling a piece 'Insect', 'Mammal' or 'Bird'. Or even perhaps 'Vertebrate'.

BTW, the thesis that names from mythology would be better known breaks down completely on this list; for almost any creature in it you can be very sure no one ever heard about it. So the question arises: why bother picking an existing historic mythological creature if  no one ever heard of it anyway? The more I think about it, the more I start to like 'Antigryph' (or 'Antigryphon', if you want): a piece that moves opposite to the Gryphon.

So, Ancaa (or Anqa) is not really another word for Gryphon (or Eagle for that matter), but a mythological creature with no parallel to a mythological creature known to English speakers. This would make sense, given that mythological creatures are made up, and different cultures are unlikely to have the same ones. It's sort of like translating the Korean Gumiho (nine-tailed fox) as a werewolf or the Japanese Oni as a demon. In that case, I don't object to using each name for a different piece. It's just a bit awkward that Betza lifted the name from the Spanish name of a piece he was using the English name for with the description given for it.

I think I have to revise the spelling in Apothecary Chess Modern then for both Griffin and Aanca!

To prevent confusion it is probably best to drop the name Aanca from English completely. We could have:

• F-then-R: Griffon
• N-then-R: Hippogriff
• W-then-B: Antigriff
• N-then-B: Monoceros

There are some other possibilities with 90-degree bends:

• W-then-R: Ultragriff, the logical extension of the series Hippogriff -> Griffon -> Ultragriff
• F-then-B: Chirogriff. (Entirely new move pattern, chimera of a Bat (Chiroptera) and a Lion, with claws at the tip of its wings.)

Logically the degenerate cases (which make a 0-degree bend) also belong in this class

• D-then-R: Ski-Rook
• A-then-B: Ski-Bishop

In the same place was described a piece called the Unicorn, moving as Knight then Bishop. Supposedly the Unicorn couldn't make a capture using its Knight move, but I'll ignore that silly rule.

Not described there is a piece which makes a one step Rook move and then continues outwards as a Bishop. For lack of a name, I'll call it the Aanca (13th century Spanish for "Gryphon"). Although the Aanca is not described, one can suppose that the same mind who conceived the Gryphon and the Unicorn probably also considered the Aanca.

I don't know if Betza ever visualized movement diagrams for these pieces. Assuming non-overlapping paths of movement, the Unicorn covers most of the same spaces as what he calls an Aanca. One of the main differences between them is that the Unicorn cannot attack any of the same spaces as a Gryphon, whereas the Aanca and the Gryphon can both attack the same spaces as a Knight. This would have been an intelligent design choice, and not a silly rule, for the following reasons:

• It makes the Gryphon and the Unicorn complementary like the Rook and Bishop are.
• It gives the Unicorn greater mobility in tight spots, which could increase it's usefulness in the midgame.
• It leaves the piece vulnerable to the Knight and keeps it from forking pieces like a Knight.

So, it seems likely that the designer did consider the "Aanca" and for the reasons I gave, rejected it in favor of the Unicorn.

Likewise, the Gryphon and what might be called the Hippogriff cover nearly the same spaces under the assumption of no overlapping paths. So that paths don't overlap, its orthogonal move could continue only in the direction of the longer part of the Knight leap. Like the Unicorn, the Hippogriff would have more mobility in tight spots, and its short-range attacking power would be crippled. If it were allowed overlapping paths, it would be a monster piece that could force checkmate against a long King, because it could attack adjacent ranks or files.

Indeed. The W-then-R and F-then-B would have overlapping paths, which could be considered a design flaw.

All these pieces have a leap followed by sliding. Of course there also are the 'hook movers' of the large Shogi variants, which have both legs of their move a slide, which makes them immensely powerful.

Given that the game had no Knights, leaving the piece vulnerable to Knight attacks would not have been important. But making its Knight move non-capturing would leave it vulnerable to all of the Gryphon's moves. It would also diminish its usefulness for defense, which would help make the game more decisive. Since the Gryphon is the most powerful piece in the game, there is less need to diminish its usefulness for defense. After all, it is the main piece that a player will use for offense if it is available. But if the Unicorn could capture with its Knight move, there would be more tempatation to use it defensively, and that could make the game more drawish. So, I think that Gryphon/Unicorn is a better choice for a game than the other two pieces used together.

Note that there is no indication whatsoever that the first step of the Grant-Acedrex N-then-B would be non-capturing in the Alfonso codex. This just states (after describing the real-life animal as a Rhino):

First, like leaps like a knight. It may remain on
that square if it wishes or may also continue to any square on the diagonal(s) of that square,
maintaining its movement in a forward direction from that square

I don't know where Betza got the idea (which he then himself rejects) that the piece could have been divergent.

Hello all. I'm reading the thread with some days late. I believe I can answer some questions. You have to know that Sonja Musser from the USA, has defended her PhD on the Libro de los Juegos, the Alfonso X's codex, few years ago. On that particular aspect of the Grant Acedrex, I worked with her analysing the Spanish text. Sonja and I speak Spanish. The language used here is a medieval Spanish but it is not a big problem to understand it because that medieval Spanish has more latin roots than the modern one. 
You can find the full text on the web and large extracts concerning our Grant Acedrex on my own website here (Spanish text AND Sonja's English translation) : http://history.chess.free.fr/acedrex.htm

The original text and a litteral translation are also given in A World Of Chess, my most recent book.

You will see that we had 2 possible interpretations for the Unicornio, which is clearly a Rhinoceros, nothing else, in the mind of the author of the 13th century. When writing A World Of Chess with my mate Rick Knowlton, we considered it again, and Rick finally convinced me that the most probable interpretation is the more natural N-then-B. I will try to update my website today.
I have studied Murray's a lot and owe a strong respect to his work. What he did is unsurpassed. No one can do something like this today. That being said, Murray's look on chess variants was not very deep. He did not have a lot of estimate for them, being more attracted by chess and its direct ancestors. He made several mistakes, especially on Grant Acedrex, but also on others like Ciccolini's chess. Having access to original sources we have corrected many small details like this in our book A World Of Chess.

To answer the last question from Fergus whether the Unicornio was capturing or not on its initial N's move: Murray affirms that cannot, but the text does not say that. The text says: 

"corre mucho desque comienca & faze ante en salto en trauiesso como Cauallo. & assi lo establecieron en este acedrex que anda el primer salto como Cauallo & depues en sosquino como la Cocatriz fata do quisiere; o que tome. E daquella casa o salta non puede tornar a tras si non yr siempre adelante." (in these times they used an u for a v)

Which means (literal translation)

it runs a lot when it begins and begins with a sideward jump like a horse, and so does it in this chess. It goes the first jump like a Horse and then goes in corner like the Crocodile does when it wants to go or take. And from that square where it jumps, it may not turn back, it shall always go forward.

In Grant Acedrex, the Crocodile moves as the MODERN Bishop. (Bishop at that time was an Alfil). 

Could the Rhino take when jumping as a Knight? Personally I believe yes, but nobody knows.

We have no more information. People from 21st century shall realise that those of the 13th century didn't have the Internet and a community of chessvariant enthusiasts to comment and discuss any tiny points of the rules :=) 

Hope this helps.