Most people approach the Needle 14ers by taking the Durango Silverton Narrow Gauge Railroad up the Animas River Gorge and hopping off at the Needleton stop. In the summer be sure to buy your tickets in advance and advise the staff that you'll be getting off at Needleton. After enjoying the scenic ride on the train and getting off, hike across the bridge and head south. The first part of the trail goes through private property so make sure to stay on the trail. After a little over a mile you'll reach the junction with the Needle Trail which heads up into Chicago Basin. The trail is easy to follow, but not too easy physically with heavy packs. Once in Chicago Basin proper after about five miles there are campsites scattered everywhere - make sure to use an already established one.
This route can also be approached from the Purgatory Trailhead. It's a longer haul, but it'll save some bucks on taking the train. The trail starts on the east side of US 550 across from the Durango Ski Area. Hike the Animas River Trail about 9 miles to the junction with the Needle Trail and proceed up that to Chicago Basin.
You can also reach Chicago Basin from deeper in the Weminuche Wilderness over Columbine Pass. Combined with other backpacking destinations and finishing with some fun in Chicago Basin before heading back out and catching the train would be one hell of a good trip.
From about 11000' in Chicago Basin the trail forks as you head east. The southern fork continues to more campsites and eventually to Columbine Pass. The left fork starts ascending slowly but still east for a little while until you reach the upper limit of where camping is allowed at around 11200'. The trail from here turns north and heads steeply up towards Twin Lakes, crossing a stream a couple of times along the way. Twin Lakes is reached at around 12500' and should take somewhere around an hour to and hour and a half from camp in Chicago Basin.
At the southernmost part of the lakes you should be able to see Eolus directly to the west. Closer in is a large rock buttress, below which you should be able to make out a trail heading west. The trail loses definition closer to the lake, so strike out west and intercept the trail as early as possible. Follow this trail up the slopes leading to a rock ampitheater until the trail turns back east and heads up a cairned rocky slab to an upper flat area framed by North Eolus and Glacier Point. The trail turns back west towards a saddle between Eolus and North Eolus. Some third class scrambling will see you to the saddle right below North Eolus. From here it is just a short few, but very enjoyable, minutes to scramble to North Eolus' summit which has terrific views north towards the Grenadiers.
Turning south you will be walking the ridge for a short distance until you reach the Catwalk, a narrow and exposed 15' section of ridge that drops steeply to both sides. It's certainly walkable, but those less enamored of exposure should have no trouble crawling. A bit further on you confront the Northeast Ridge itself, which rises much more steeply to the summit. The ridge is fourth class and will see you to the summit more directly and quickly, but if you prefer to keep the difficulties to third class traverse left around the ridge to the East Face. The way up the East Face winds its way up gradually gaining elevation until the ridge south of the summit is attained. Pay close attention for the cairns which indicate the path of least resistance, although there are cairns all over the place. Once on the ridge, head north a short distance to the summit of Eolus and fabulous views of Pigeon, Monitor, Twin Lakes, North Eolus, Columbine Pass, Sunlight, Windom, and many more!! The descent is a reverse of the ascent - just be careful of not slipping on the more exposed sections!
A helmet might be a good idea, as well as a rope and simple climbing gear for those more timid. The scrambling on the East Face is relatively simple, but if you decide to tackle the ridge proper there is more difficult scrambling and exposure.
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