Discerning the Transmundane

Background
"I met Septimus Signus, a brilliant but mad scholar living in the northern ice fields. He's directed me to a dwarven observatory that supposedly houses an Elder Scroll. He doesn't care about the Scroll itself, but wants me to use some kind of machine there to inscribe a dwarven lexicon with the Scroll's knowledge and bring it to him."

Walkthrough
The hermit Septimus Signus has devoted his life to the Elder Scrolls. Septimus is located in Septimus Signus's Outpost, an ice cave north of Winterhold. Travel to Septimus, who'll give you a Lexicon and sends you to Blackreach, an underground dwarven city, which you enter through Alftand. Follow the quest-marker to the Tower of Mzark.

Inside the tower, you will encounter a sphere which you will need to go through a puzzle where you need to push buttons in order to open. Activate the Lexicon Reptacle to the right of the buttons to gain access to the two buttons to the right. To decipher the code, press the second button from the right until the Lexicon opens and the third button lights up. Press the third button until you see the overhead lenses directing light onto the sphere, and the fourth button lights up. Press the last button and retrieve the scroll. Now pick up the transcribed Lexicon (that's the cube you put in the mechanism) from the reptacle and then procceed to pick up the Elder Scroll.

Harvesting Blood
Return to Septimus, who will (provided you are at least Level 15) now ask you to gather some blood in order to make a dwarven blood replica to open the door to the dwarven artifact in his outpost. You need to gather the blood of the following races: To harvest the blood, you need to loot the corpse of someone from each race. A good place to get this is Liar's Retreat, which has all of the blood types, except the High Elf, which is most easily obtained in Pinewatch. (Enter the hidden underground tunnel by pressing a button above the table in the basement of Pinewatch, and once you enter the first room in the tunnel, the High Elf should be directly to your left.) Another easy way of acquiring Altmer blood is by visiting the Thalmor Embassy and killing the first Thalmor you see, who will be standing just inside the courtyard of the Embassy. He's alone, and killing him should raise no alarm.
 * Orc
 * Falmer
 * Dark Elf
 * Wood Elf
 * High Elf

Also, if you have Beast Form, you can kill any NPC of each races in any city without raising a Bounty on your head. Simply kill them while in beast form, run away / exit combat, wait for your beast form to wear off and run back to the corpses to harvest the blood needed.

Possible Bug
After returning the Lexicon, Septimus will not ask you to retrieve the blood of each race; he will say that he will contact you later, but no new quest objective is added. The conversation option required to be able to continue in the quest will never appear in Septimus' dialogue options. This glitch may be caused by finding and returning the Lexicon to Septimus before the player reaches level 15. Reloading and then advancing to level 15 before giving Septimus the Lexicon seems to enable the quest to progress normally. (360)(PS3)(PC)

This can also be fixed by just waiting a few days (after reaching 15), a courier will bring you a note from septimus telling you to return to him. (360) Although on PS3 waiting a couple days does not work there does not seem to be a fix for it. In order to get the new quest prompt to send you to Blackreach, you have to use the dialogue option to ask Septimus where Blackreach is. Sometimes this works, and sometimes the system (PS3) will register the question as already having been asked. Wait for 24 hours, and it will reset the dialogue cycle.

Occasionally (PS3) this will not work. If this is the case and you have waited 24 hours and the dialogue box has not reset, then you must reset your file and start over.

Hermaeus Mora's Intervention
Upon trying to leave the iceberg, Hermaeus Mora, Daedric Prince of the unknown, confronts you (actually blocks the exit) and tells you that you have been doing well. He tells you that as soon as the lock box is open, Septimus Signus' usefulness will be over, and that the Dragonborn should take his place. You have the option to agree or disagree.

Disagree
You tell the wretched Abyss you will never be their servant, and its response is "you shall not evade me forever"; then it disappears.

Agree
He then asks you to speak with him once the box has been opened, and all will be revealed.

Either you agree to be his follower or not; this will not have any (at least not noticeable) consequences or differences after finishing the quest.

Suggested Sources of Blood
The following are suggested sources of blood which you may turn to if you can't find any on your own.

The best place to gather blood would be at Liar's Retreat (lies south-south-west of Dragon Bridge) or at Chillwind Depths Just east of Liar's Retreat. The cave has many falmer and the corpses of all races. The easiest place to find a high elf is the corpse in the spear pit out the back entrance of Halted Stream Camp. Another good spot to find a high elf is north-west of Helgen. Between Pinewatch and Anise's Cabin you'll find a hostile Novice Conjurer, and a Shrine of Talos with a dead Thalmor wizard on the mountainside directly South of Anise's Cabin.

Alternatively you can obtain the high elf blood by traveling to Solitude, then give the store Radiant Raiment a tiny visit. This store is run by two high elves, Taarie and Endarie. You should kill Endarie as she doesn't provide you with any quests. Taarie will still be there to run the store selling the same items Endarie did.

Helgen, now infested with bandits of myriads of races in an exellent source of blood. The player should be able to get the blood of all other races other than Falmer blood. Killing the bandits can be done without any repercussions. If you are not able to find the all the blood you need from the outside ruins of Helgen head on to the inside ruins where most of the tutorial mission took place.

The Thalmor Embassy is also a very good location to find several High Elves. You can lock pick your way into the guard barracks and harvest the blood of an unsuspecting sleeper. (When I played through the mission 'Diplomatic Immunity', I found High Elf, Wood Elf, and Dark Elf blood.)

An even better source of High Elf blood is at Northwatch Keep, along the shoreline Northeast of Mor Khazgur or Northwest of Volskygge.

Between the bandit squatters and falmer inhabitants, Mzinchaleft contains all blood types except High Elf.

High Elf and Dark Elf Blood can be found at Cragwallow Slope, which is slightly southeast of Kynesgrove. It is a cavern filled with elven Conjurers. Or Ilinalta's Deep which has elven necromancers. So at least one of the two types of required Elf blood can be found in either locations.

If you don't feel like killing off named NPCs or are trying to avoid criminal acts, hostile high elves can also be found at Uttering Hills Cave in the form of the Summerset Shadows or at the Thalmor Embassy. There is also a High Elf named Nurelion in The White Phial in Windhelm that dies of natural causes during the Repairing the Phial quest.

A hostile Wood Elf can normally be found among the bandits at Silent Moons Camp. There is a Wood Elf in Riverwood named Faendal who can be used for blood. Note: Doing this will not allow you to complete his associated quests.

Two other Wood Elves can be found in Whiterun and are the owners of The Drunken Huntsman. They are Anoriath and Elrindir. Attacking them can lead to the guards being called. There is also a Wood elf corpse in Sunderstone Gorge next to the Word of Power for Fire Breath. You will also encounter the other 2 types of elves as enemies here (jr).

If for some reason the "Dark Elf" was not in Liar's Retreat (as stated above) then go to Riften. There is a Dark Elf in the market district that is female. She sells Harvested goods so just take her out, then pay off the guard 350 gold and you're on your way.

Another way to obtain the Dark Elf blood is to visit Drelas' Cottage. Once you enter Drelas will become hostile immediately so be prepared. This is a good way to obtain the Dunmer blood if you do not want to deal with a bounty.

If you have enough money, it is possible to go to towns and murder passersby for the necessary blood, then simply pay your bounty. This is made easier if part of the Thieve's Guild. The best places to find this necessary blood is Mzinchaleft (Falmer), Markarth (Orc and Altmer), and Windhelm (Others).

The best thing to do after giving the lexicon to Septimus is to simply go back to Alftand. You can find all the blood types except for dunmer blood there. If you cleared the dungeon then it should be safe enough to run through it again (unless you sneaked past it). The body of the woodelf is found along a small passage way upon the Alftand Cathedral. You will see a small opening with a puddle of blood. Go through the small passage and you will find the woodelf body. The orc blood can be found on the corpse of a female orc halfway through the spiraling path further into Alftand. Continuing through, you will find Falmer right after discovering the orc. Kill them (if you already went through the dungeon corpses might already be gone). Finally, as you enter the second to last part of Alftand, you will find a High Elf corpse. There are also three leveled Falmer enemies here and a Falmer corpse that will always be there no matter what. As stated above, Drelsa's Cottage is the best way to find Dunmer blood without a bounty.

Quest Conclusion
After you've gathered the blood, return to Septimus. He will open the dwemer box using the blood you gathered but as he enters inside and finds the pedestal on which the Oghma Infinium resides he is killed by Hermaeus Mora by suddenly disintegratrating him into a pile of ash.

Step inside and take the book. As you leave the chamber the Wrectched Abyss appears again as a manifestation of Hermaeus Mora. You can decide to work together with him as his Champion or to disagree with any of his plans. This choice doesn't affect your actions in any way.

Once you open the book to read, you have three choices of skill upgrade (or choose "Do not read").

Bugs

 * When Septimus finds the book, Hermaeus Mora might not kill him and the quest will get stuck. This will most likely happen if you stay really close to septimus, try to talk to him and read the book before he finishes speaking (PC). However, you can just kill Septimus yourself and the quest will then continue as it should. (PC)
 * After Septimus is killed, if you leave the iceberg without taking or reading the book and continue on other quests once you return, the dwemer box may be locked again with no way to unlock it and retrieve the book inside, even if you still have the essence extractor, Lexicon etc. (360)
 * After completing the quest, the lexicon and/or essesence extractor are still considered quest items and are unable to be removed from your inventory. (PS3/Xbox 360)
 * This may not be a bug after all as the lexicon and the sphere are used to open certain Dwarven doors and puzzles.